Research ArticleSecond-Order Linear Differential Equations with Solutions inAnalytic Function Spaces

Jianren Long 12 Yu Sun1 Shimei Zhang1 and Guangming Hu3

1School of Mathematical Science Guizhou Normal University Guiyang 550001 China2School of Computer Science and School of Science Beijing University of Posts and Telecommunications Beijing 100876 China3School of Sciences Jinling Institute of Technology Nanjing 211169 China

Correspondence should be addressed to Jianren Long longjianren2004163com

Received 13 November 2018 Accepted 10 January 2019 Published 12 February 2019

Academic Editor Ruhan Zhao

Copyright copy 2019 Jianren Long et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This research is concerned with second-order linear differential equation 11989110158401015840 +119860(119911)119891 = 0where119860(119911) is an analytic function in theunit disc On the one hand some sufficient conditions for the solutions to be in 120572-Bloch (little 120572-Bloch) space are found by usingexponential type weighted Bergman reproducing kernel formula On the other hand we find also some sufficient conditions forthe solutions to be in analytic Morrey (little analytic Morrey) space by using the representation formula

1 Introduction

We shall assume that the reader is familiar with the defini-tions of classical function spaces for example Hardy spaceBergman space and Bloch space One of main objectivesin the research of complex linear differential equations withanalytic coefficients in unit disc is to consider the relationshipbetween the growth of coefficients and the growth of solu-tions Many results concerning fast growing solutions havebeen obtained by Nevanlinna and Wiman-Valiron theoriesfor example see [1ndash5] and reference therein However itis very difficult to study the slowly growing solutions ofcomplex linear differential equations by using NevanlinnaandWiman-Valiron theories hence the different approachesare employed for example Heroldrsquos comparison theorem [6]Gronwallrsquos lemma [7] Picardrsquos successive approximations [8]and some methods based on Carleson measures [9] Therearemany results concerning the slowly growing solutions thathave been obtained for example see [6 7 9ndash11] and referencetherein

In [9] Pommerenke considered the second-order equa-tion

11989110158401015840 + 119860 (119911) 119891 = 0 (1)

where 119860(119911) is an analytic function in the unit disc D =119911 |119911| lt 1 and obtained a sharp sufficient conditionsfor 119860(119911) which guarantee all solutions of (1) are in theclassical Hardy space 1198672(D) The coefficient condition wasgiven in terms of Carleson measures by Greenrsquos formula andCarlesonrsquos theorem for Hardy space The leading methodof this approach has been extended to study for examplesolutions in the Hardy space and Bergman space [6] andDirichlet type space [10]

Recently Grohn-Huusko-Rattya [12] found sufficientconditions for the coefficient 119860(119911) guaranteeing all solutionsof (1) are in Bloch (little Bloch) space by taking advantageof the reproducing formula in small weighted Bergmanspace [13] Motivated from [12] now we use the reproducingformula in Bergman space with exponential type weightsto get the conditions for the coefficient 119860(119911) such that allsolutions of (1) are in 120572-Bloch (little 120572-Bloch) space In thesame paper they obtained also some sufficient conditionson the coefficient 119860(119911) which guarantee all solutions of (1)belong to BMOA (VMOA) space in which some propertiesof Bloch space were used It follows from [14ndash16] thatBMOA space is a special kind of analytic Morrey spaceand the analytic Morrey space and Bloch space are differentTherefore we obtain also some sufficient conditions on 119860(119911)

HindawiJournal of Function SpacesVolume 2019 Article ID 8619104 9 pageshttpsdoiorg10115520198619104

2 Journal of Function Spaces

which guarantee all solutions of (1) to be in analytic Morrey(little analytic Morrey) space by using different ways in [12]The definition of BMOA space and analytic Morrey space arerecalled in Section 2 below

Note If 119860 ≲ 119861 then there exists a positive constant 119862 suchthat 119860 le 119862119861 Similarly if 119860 ≳ 119861 then there exists a positiveconstant119862 such that119860 ge 119862119861 If119860 and119861 satisfy119860 ≳ 119861 and119860 ≲119861 then we say that A and B are comparable by 119860 asymp 119861 Notethat 120590119886(119911) = (119886 minus 119911)(1 minus 119886119911) is the Mobius transformation ofD for 119886 119911 isin D and 119889119898(119911) denotes the normalized Lebesguearea measure

The paper is organized as follows some definitions ofrelated function spaces are recalled in Section 2 Some resultsin which all solutions of (1) are in 120572-Bloch spaces arediscussed in Section 3 We obtain some results in which allsolutions of (1) are in analytic Morrey spaces in Section 4

2 Some Related FunctionSpaces and Notations

Let 120597D be the boundary of D and A(D) be the space ofanalytic functions in D For 1 le 119901 lt infin Hardy space119867119901(D)consists of 119891 isin A(D) with

10038171003817100381710038171198911003817100381710038171003817119901119867119901 = sup0lt119903lt1

12120587 int2120587

0

10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816119901 119889120579 lt infin (2)

For a subarc 119868 sub 120597D the length of 119868 is defined as

|119868| = 12120587 int119868 10038161003816100381610038161198891205771003816100381610038161003816 (3)

and let

119878 (119868) = 119903120577 isin D 1 minus |119868| le 119903 lt 1 120577 isin 119868 (4)

denote the Carleson square in DDenote

119891119868 = 1|119868| int119868 119891 (120577)100381610038161003816100381611988912057710038161003816100381610038162120587 (5)

as the average of 119891 isin A(D) over 119868 BMOA space consists ofthose functions 119891 isin 1198672(D) such that

10038171003817100381710038171198911003817100381710038171003817BMOA = (sup119868sub120597D

1|119868| int119868 1003816100381610038161003816119891 (120577) minus 11989111986810038161003816100381610038162100381610038161003816100381611988912057710038161003816100381610038162120587 )12 lt infin (6)

VMOA space consists of those functions 119891 isin BMOA suchthat

lim|119868|997888rarr0

1|119868| int119868 1003816100381610038161003816119891 (120577) minus 11989111986810038161003816100381610038162100381610038161003816100381611988912057710038161003816100381610038162120587 = 0 (7)

Morrey spaces were introduced in the 1930s [17] inconnection to partial differential equations and were subse-quently studied as function spaces in harmonic analysis onEuclidean spaces The analytic Morrey space L2120582(D) (0 le120582 le 1) was introduced recently by Wu and Xie [15] and then

many researchers pay attention to the spaces for example see[16 18 19] and reference therein The analytic Morrey spaceL2120582(D) consists of those functions 119891 isin 1198672(D) such that

10038171003817100381710038171198911003817100381710038171003817L2120582 = sup119868sub120597D

( 1|119868|120582 int119868 1003816100381610038161003816119891 (120577) minus 11989111986810038161003816100381610038162100381610038161003816100381611988912057710038161003816100381610038162120587 )12 lt infin (8)

Little analytic Morrey space L21205820 (D) consists of those func-

tions 119891 isin L2120582(D) andlim

|119868|997888rarr0

1|119868|120582 int119868 1003816100381610038161003816119891 (120577) minus 11989111986810038161003816100381610038162100381610038161003816100381611988912057710038161003816100381610038162120587 = 0 (9)

Clearly for 120582 = 0 or 120582 = 1 L2120582(D) reduces to 1198672(D) andBMOA respectively hence the case of 0 lt 120582 lt 1 is markedand then BMOAsub L2120582(D) sub 1198672(D)

The following lemma gives some equivalent conditions ofL2120582(D) by [19 Theorem 31] or [20 Theorem 321]

Lemma 1 Suppose that 0 lt 120582 le 1 and 119891 isin A(D) Then thefollowing statements are equivalent

(i) 119891 isin L2120582(D)(ii) sup119886isinD(1 minus |119886|2)1minus120582 int

D|1198911015840(119911)|2(1 minus |120590119886(119911)|2)119889119898(119911) ltinfin

(iii) sup119868sub120597D(1|119868|120582) int119878(119868) |1198911015840(119911)|2(1 minus |119911|2)119889119898(119911) lt infin

(iv) sup119886isinD(1 minus |119886|2)(1minus120582)2119891 ∘ 120590119886 minus 119891(119886)1198672 lt infin

From Lemma 1 we see the following

Lemma 2 Let 119891 and 120582 be defined as Lemma 1 Then 119891 isinL2120582

0 (D) if and only iflim

|119886|997888rarr1minus(1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0 (10)

or

lim|119886|997888rarr1minus

(1 minus |119886|2)(1minus120582)2 1003817100381710038171003817119891 ∘ 120590119886 minus 119891 (119886)10038171003817100381710038171198672 = 0 (11)

Here we use the reproducing formula in Bergman spacewith exponential type weights to study (1) To this end let0 lt 119901 lt infin the exponential type weighted Bergman space119860119875(120596120574120572) is the space of functions 119891 isin A(D) such that

10038171003817100381710038171198911003817100381710038171003817119901119860119875(120596120574120572) fl intD

1003816100381610038161003816119891 (119911)1003816100381610038161003816119901 120596120574120572 (|119911|) 119889119898 (119911) lt infin (12)

where 120596120574120572 is the exponential type weight as

120596120574120572 (120588) = (1 minus 120588)120574 exp( minus119888(1 minus 120588)120572) 120574 ge 0 119888 gt 0 120572 gt 0

(13)

In [21] it is proved that the point evaluation 119871120577(119891(119911)) = 119891(120577)is bounded linear function on 119860119875(120596120574120572) Hence there exists a

Journal of Function Spaces 3

reproducing kernel 119870120577 isin 119860119875(120596120574120572) with 1198701205771198602(120596120574120572) = 119871120577such that

119891 (120577) = 119871120577 (119891 (119911)) fl intD

119891 (119911)119870120577 (119911)120596120574120572 (|119911|) 119889119898 (119911) 119891 isin 119860119875 (120596120574120572)

(14)

where

119870120577 (119911) = infinsum119899=0

(119911120577)1198992 int1

01205882119899+1120596120574120572 (120588) 119889120588 (15)

Finally we recall the definition of 120572-Bloch space B120572which can be found in [22] Let 120572 isin (0infin) it is defined as

B120572

fl ℎ isin A (D) ℎB120572 fl sup119911isinD

10038161003816100381610038161003816ℎ1015840 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120572 lt infin (16)

Little 120572-Bloch space B1205720 is the subspace of B120572 consisting of

functions ℎ withlim

|119911|997888rarr1minus

10038161003816100381610038161003816ℎ1015840 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120572 = 0 (17)

ClearlyB120572 (B1205720 ) is the Bloch (little Bloch) space for 120572 = 1

3 Sufficient Conditions ofSolutions in B120572 (B120572

0)

In the section we consider the estimation of coefficientsimilarly to Section 7 in [12] in which the exponential typeweighted Bergman reproducing kernel formula is used Somesufficient conditions on 119860(119911) guaranteeing all solutions of (1)to be inB120572 (B120572

0 ) are obtained by these estimates as follows

eorem 3 Let 120596120574120572(120588) be an exponential type weight If

lim120588997888rarr1minus

sup119911isinD

(1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

(18)

is sufficiently small then all solutions of (1) belong toB2(1+120572)

eorem 4 Let 120596120574120572(120588) be an exponential type weight Sup-pose that the condition of Theorem 3 is satisfied and

lim|119911|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816) = 0

(19)

Then all solutions of (1) belong toB2(1+120572)0

In order to proveTheorems 3 and 4 we need the followinglemma

Lemma 5 (see [21]) If 119891 119892 isin 1198602(120596120574120572) thenintD

119891 (119911) 119892 (119911)120596120574120572 (|119911|) 119889119898 (119911)= int

D

1198911015840 (119911) 1198921015840 (119911) (1 minus |119911|)2(1+120572) 120596120574120572 (|119911|) 119889119898 (119911)+ 119891 (0) 119892 (0)

(20)

Proof of Theorem 3 Let 119891 be any solution of (1) then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120577) 119860 (120588120577) 119889120577 + 1198911015840

120588 (0) 119911 isin D (21)

By the reproducing formula and Fubinirsquos theorem

1198911015840120588 (119911)= minusint119911

0(int

D

119891120588 (120578)119870120577 (120578)120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) 119889119898 (120578)) 1205882119860 (120588120577) 119889120577+ 1198911015840

120588 (0)= minusint

D

119891120588 (120578) 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

0119870120577 (120578)1205882119860 (120588120577) 119889120577) 119889119898 (120578)

+ 1198911015840120588 (0) 119911 isin D

(22)

Applying Lemma 5

1198911015840120588 (119911) = minusint

D

1198911015840120588 (120578) (1 minus 10038161003816100381610038161205781003816100381610038161003816)2(1+120572) 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816)

sdot (int119911

01198701015840

120577(120578)1205882119860 (120588120577) 119889120577) 119889119898 (120578) + 119891120588 (0)

sdot (int119911

0

1205882119860 (120588120577)2 int1

0120588120596120574120572 (120588) 119889120588119889120577) + 1198911015840

120588 (0) (23)

Therefore1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) ≲ 1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) sup

119911isinD(1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

+ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 sup119911isinD

(1 minus |119911|)2(1+120572)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816int119911

0

119860 (120588120577)2 int1

0120588120596120574120572 (120588) 119889120588119889120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161198911015840120588 (0)10038161003816100381610038161003816 119911 isin D

(24)

This implies that

1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) (1 minus 119862 sup119911isinD

(1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

≲ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 sup119911isinD

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 + 100381610038161003816100381610038161198911015840

120588 (0)10038161003816100381610038161003816 (25)

4 Journal of Function Spaces

where 119862 is a positive constant and 0 lt 120588 lt 1 It follows fromFubinirsquos theorem and the reproducing formula that

sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

= sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0(int

D

120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) 1198701015840120577(120578)119889119898 (120578))119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816)

≳ sup119911isinD

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(26)

Therefore we deduce 119891 isin B2(1+120572) by the above formula and(25) and letting 120588 997888rarr 1minusProof ofTheorem 4 ByTheorem 3 we know that119891 isin B2(1+120572)Fubinirsquos theorem and the reproducing formula yield

lim|119911|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

≳ lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(27)

The condition of Theorem 4 implies

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 = 0 (28)

By using the similar reason as in the proof of Theorem 3 wehave

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 ≲ 1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) (1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

+ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 (1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 + (1 minus |119911|)2(1+120572)

sdot 100381610038161003816100381610038161198911015840120588 (0)10038161003816100381610038161003816 119911 isin D

(29)

Then by the condition of Theorem 4

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 = 0 (30)

Therefore the assertion follows

4 Sufficient Conditions ofSolutions in L2120582(D) (L2120582

0 (D))In the section some sufficient conditions on 119860(119911) whichguarantee all solutions of (1) belong to analytic Morrey space

are obtained by using two ideas On the one hand a sufficientcondition on119860(119911) guaranteeing that all solutions of (1) belongto BMOA is obtained in [12 Theorem 3] here we study thecondition on 119860(119911) which guarantee that all solutions of (1)are in analytic Morrey space by using similar idea in [12Theorem3]On the other hand a sufficient condition on119860(119911)guaranteeing that all solutions of (1) are in analytic Morreyspace is shown by using representation formula

It is also known that BMOA space is a subspace of Blochspace and then the properties of Bloch space were used in theproof of [12 Theorem 3] However it was proved that analyticMorrey space and Bloch space are different in [16]Thereforewe need different methods from Section 6 of [12] to deal withthe case of analytic Morrey space the following results areproved

eorem 6 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatsup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (31)

is sufficiently small and

sup119911isinD

|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2 lt 3 minus 1205822 (32)

Then all solutions of (1) are inL2120582(D)eorem 7 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatthe conditions of Theorem 6 are satisfied and

lim|119886|997888rarr1minus

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0 (33)

and

lim|119911|997888rarr1minus

(|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2) = 0 (34)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 6 and 7 we recall thedefinition of Carleson measure For a non-negative measure120583 on D it is called a Carleson measure if

10038171003817100381710038171205831003817100381710038171003817 š sup119868sub120597D

(120583 (119878 (119868))|119868| )12 lt infin (35)

Moreover if in addition

lim|119868|997888rarr0

120583 (119878 (119868))|119868| = 0 (36)

then 120583 is called a compact Carleson measureFor the proof of Theorems 6 and 7 the following lemmas

are needed

Lemma 8 (sse [14]) Suppose that 120583 is a Carleson measureThen

10038171003817100381710038171205831003817100381710038171003817 asymp sup119886isinD

intD

100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889120583 (119911) (37)

Journal of Function Spaces 5

Lemma 9 (see [23]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) thenintD

1003816100381610038161003816119891 (119911)1003816100381610038161003816119901 (1 minus |119911|2)119902 119889119898 (119911)≲ (1003816100381610038161003816119891 (0)1003816100381610038161003816119901 + int

D

100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119901+119902 119889119898 (119911)) (38)

Lemma 10 (see [24]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) then10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+2≲ int

D

10038161003816100381610038161003816119891(119899) (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119902 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (39)

Lemma 11 (see [22]) Suppose that 0 lt 119901 119902 lt infin Then for119891 isin A(D)sup119886isinD

10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902 (40)

and

sup119886isinD

10038161003816100381610038161003816119891(119899+1) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+119901 (41)

are comparable

Lemma 12 Let 0 lt 120582 lt 1 and 119891 isin L2120582 Then

100381710038171003817100381711989110038171003817100381710038172L2120582 ≲ (sup119886isinD

(100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)3minus120582)+ sup

119886isinD((1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119886 (119911)210038161003816100381610038161003816) 119889119898 (119911))) (42)

Proof An auxiliary function is constructed as follows

119892 (119911) = 1198911015840 (120590119886 (119911)) (1 minus |119886|2)(1 minus 119886119911)2 (43)

By Lemmas 1 and 9100381710038171003817100381711989110038171003817100381710038172L2120582 asymp (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 ∘ 120590119886 (119911)10038161003816100381610038162)sdot 100381610038161003816100381610038161205901015840119886 (119911)100381610038161003816100381610038162 119889119898 (119911) = (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2 119889119898 (119911)1 minus |119911|2 = (1minus |119886|2)1minus120582 int

D

1003816100381610038161003816119892 (119911)10038161003816100381610038162 (1 minus |119911|2) 119889119898 (119911) ≲ (1003816100381610038161003816119892 (0)10038161003816100381610038162sdot (1 minus |119886|2)1minus120582 + (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911))

(44)

where |119892(0)|2 = |1198911015840(119886)|2(1 minus |119886|2)2

Note that

(1 minus |119886|2)1minus120582 intD

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911) le 1198681 + 1198682 (45)

where

1198681 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161003816(1198911015840 (120590119886 (119911)))10158401003816100381610038161003816100381610038162|1 minus 119886119911|4 (1 minus |119886|2)2 (1 minus |119911|2)3 119889119898 (119911)(46)

and

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus |119886|2)2|1 minus 119886119911|6 (1 minus |119911|2)3 119889119898 (119911) (47)

Clearly

1198681 ≲ (1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)41 minus |119911|2 119889119898 (119911)

= (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(48)

It follows from Lemmas 10 and 11 that

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2|1 minus 119886119911|2 (1 minus |119911|2) 119889119898 (119911)le (1 minus |119886|2)1minus120582 sup

119886isinD100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)2

sdot intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911) ≲ (1 minus |119886|2)1minus120582

sdot sup119886isinD

1003816100381610038161003816100381611989110158401015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)4intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)≲ (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)sdot int

D

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)

(49)

It is easy to get that sup119886isinD intD((1 minus |119911|2)|1 minus 119886119911|2)119889119898(119911) isfinite Then the lemma is completely proved

6 Journal of Function Spaces

Lemma 13 (see [25]) Suppose that 120583 is a positive Borelmeasure and 0 lt 119901 lt infin Then 120583 is a Carleson measure if andonly if 119867119901(D) sub 119871119901(119889120583) Namely if there exists a continuousincluding mapping 119894 119867119901(D) 997888rarr 119871119901(119889120583) then

10038171003817100381710038171198911003817100381710038171003817119871119901(119889120583) ≲ 10038171003817100381710038171205831003817100381710038171003817 10038171003817100381710038171198911003817100381710038171003817119867119901(D) (50)

Lemma 14 (see [18]) Let 0 lt 120582 lt 1 If 119891 isin L2120582 then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817L2120582

(1 minus |119911|2)(1minus120582)2 119911 isin D (51)

Proof of Theorem 6 We can obtain that all solutions 119891 of (1)belong toB(3minus120582)2 by [11 Corollary 4] and (32) It follows fromLemma 12 that

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 ≲ sup119886isinD

((1 minus |119886|2)3minus120582 100381610038161003816100381610038161198911015840 (120588119911)100381610038161003816100381610038162 1205882+ (1 minus |119886|2)1minus120582sdot int

D

1205884 1003816100381610038161003816100381611989110158401015840 (120588119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911))≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + sup

119886isinD(1 minus |119886|2)1minus120582 sdot int

D

10038161003816100381610038161003816119891120588 (119911) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)+ sup

119886isinD(1 minus |119886|2)1minus120582 10038161003816100381610038161003816119891120588 (119886)100381610038161003816100381610038162 int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + 1198691+ 1198692

(52)

where 119891120588(119911) = 119891(120588119911) By Lemmas 1 8 and 13

1198691 ≲ sup119886isinD

(1 minus |119886|2)1minus120582 intD

10038161003816100381610038161003816119891120588 (120590 (119911)) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 sdot (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889119898 (119911)≲ sup

119886isinD(1 minus |119886|2)1minus120582 (10038171003817100381710038171003817119891120588 (120590 (119911)) minus 119891120588 (119886)1003817100381710038171003817100381721198672

sdot sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119887

∘ 120590minus1119886 (119911)100381610038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119888isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119888 (119911)10038161003816100381610038162) 119889119898 (119911)

(53)

where 119889120583(119911) = |119860(120588120590119886(119911))|2(1 minus |120590119886(119911)|2)3|1205901015840119886(119911)|119889119898(119911) is aCarleson measure

It follows from Lemma 14 that

1198692 ≲ sup119886isinD

(1 minus |119886|2)1minus12058210038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

(1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(54)

Denote

119869 (119886 120588) = intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) 0 lt 120588 lt 1

(55)

For |119886| le 12 the estimate is trivial Let 12 le |119886| le 1(2minus120588)Since |1 minus 119886119911| le 2|1 minus 119886119911120588| for |119911| le 120588119869 (119886 120588)

= intD(0120588)

|119860 (119911)|2 (1 minus 10038161003816100381610038161003816100381610038161003816 119911120588100381610038161003816100381610038161003816100381610038162)2 1 minus |119886|210038161003816100381610038161 minus 119886 (119911120588)1003816100381610038161003816

119889119898 (119911)1205882le 41205882 intD |119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)le 16int

D|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(56)

for any 12 lt 120588 lt 1 Let 1(2 minus 120588) le |119886| le 1 Now119869 (119886 120588) le sup

119886isinD|119860 (119911)|2 (1 minus |119911|2)4

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(57)

Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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Journal of Function Spaces 3

reproducing kernel 119870120577 isin 119860119875(120596120574120572) with 1198701205771198602(120596120574120572) = 119871120577such that

119891 (120577) = 119871120577 (119891 (119911)) fl intD

119891 (119911)119870120577 (119911)120596120574120572 (|119911|) 119889119898 (119911) 119891 isin 119860119875 (120596120574120572)

(14)

where

119870120577 (119911) = infinsum119899=0

(119911120577)1198992 int1

01205882119899+1120596120574120572 (120588) 119889120588 (15)

Finally we recall the definition of 120572-Bloch space B120572which can be found in [22] Let 120572 isin (0infin) it is defined as

B120572

fl ℎ isin A (D) ℎB120572 fl sup119911isinD

10038161003816100381610038161003816ℎ1015840 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120572 lt infin (16)

Little 120572-Bloch space B1205720 is the subspace of B120572 consisting of

functions ℎ withlim

|119911|997888rarr1minus

10038161003816100381610038161003816ℎ1015840 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120572 = 0 (17)

ClearlyB120572 (B1205720 ) is the Bloch (little Bloch) space for 120572 = 1

3 Sufficient Conditions ofSolutions in B120572 (B120572

0)

In the section we consider the estimation of coefficientsimilarly to Section 7 in [12] in which the exponential typeweighted Bergman reproducing kernel formula is used Somesufficient conditions on 119860(119911) guaranteeing all solutions of (1)to be inB120572 (B120572

0 ) are obtained by these estimates as follows

eorem 3 Let 120596120574120572(120588) be an exponential type weight If

lim120588997888rarr1minus

sup119911isinD

(1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

(18)

is sufficiently small then all solutions of (1) belong toB2(1+120572)

eorem 4 Let 120596120574120572(120588) be an exponential type weight Sup-pose that the condition of Theorem 3 is satisfied and

lim|119911|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816) = 0

(19)

Then all solutions of (1) belong toB2(1+120572)0

In order to proveTheorems 3 and 4 we need the followinglemma

Lemma 5 (see [21]) If 119891 119892 isin 1198602(120596120574120572) thenintD

119891 (119911) 119892 (119911)120596120574120572 (|119911|) 119889119898 (119911)= int

D

1198911015840 (119911) 1198921015840 (119911) (1 minus |119911|)2(1+120572) 120596120574120572 (|119911|) 119889119898 (119911)+ 119891 (0) 119892 (0)

(20)

Proof of Theorem 3 Let 119891 be any solution of (1) then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120577) 119860 (120588120577) 119889120577 + 1198911015840

120588 (0) 119911 isin D (21)

By the reproducing formula and Fubinirsquos theorem

1198911015840120588 (119911)= minusint119911

0(int

D

119891120588 (120578)119870120577 (120578)120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) 119889119898 (120578)) 1205882119860 (120588120577) 119889120577+ 1198911015840

120588 (0)= minusint

D

119891120588 (120578) 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

0119870120577 (120578)1205882119860 (120588120577) 119889120577) 119889119898 (120578)

+ 1198911015840120588 (0) 119911 isin D

(22)

Applying Lemma 5

1198911015840120588 (119911) = minusint

D

1198911015840120588 (120578) (1 minus 10038161003816100381610038161205781003816100381610038161003816)2(1+120572) 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816)

sdot (int119911

01198701015840

120577(120578)1205882119860 (120588120577) 119889120577) 119889119898 (120578) + 119891120588 (0)

sdot (int119911

0

1205882119860 (120588120577)2 int1

0120588120596120574120572 (120588) 119889120588119889120577) + 1198911015840

120588 (0) (23)

Therefore1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) ≲ 1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) sup

119911isinD(1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

+ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 sup119911isinD

(1 minus |119911|)2(1+120572)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816int119911

0

119860 (120588120577)2 int1

0120588120596120574120572 (120588) 119889120588119889120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161198911015840120588 (0)10038161003816100381610038161003816 119911 isin D

(24)

This implies that

1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) (1 minus 119862 sup119911isinD

(1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

≲ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 sup119911isinD

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 + 100381610038161003816100381610038161198911015840

120588 (0)10038161003816100381610038161003816 (25)

4 Journal of Function Spaces

where 119862 is a positive constant and 0 lt 120588 lt 1 It follows fromFubinirsquos theorem and the reproducing formula that

sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

= sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0(int

D

120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) 1198701015840120577(120578)119889119898 (120578))119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816)

≳ sup119911isinD

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(26)

Therefore we deduce 119891 isin B2(1+120572) by the above formula and(25) and letting 120588 997888rarr 1minusProof ofTheorem 4 ByTheorem 3 we know that119891 isin B2(1+120572)Fubinirsquos theorem and the reproducing formula yield

lim|119911|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

≳ lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(27)

The condition of Theorem 4 implies

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 = 0 (28)

By using the similar reason as in the proof of Theorem 3 wehave

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 ≲ 1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) (1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

+ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 (1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 + (1 minus |119911|)2(1+120572)

sdot 100381610038161003816100381610038161198911015840120588 (0)10038161003816100381610038161003816 119911 isin D

(29)

Then by the condition of Theorem 4

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 = 0 (30)

Therefore the assertion follows

4 Sufficient Conditions ofSolutions in L2120582(D) (L2120582

0 (D))In the section some sufficient conditions on 119860(119911) whichguarantee all solutions of (1) belong to analytic Morrey space

are obtained by using two ideas On the one hand a sufficientcondition on119860(119911) guaranteeing that all solutions of (1) belongto BMOA is obtained in [12 Theorem 3] here we study thecondition on 119860(119911) which guarantee that all solutions of (1)are in analytic Morrey space by using similar idea in [12Theorem3]On the other hand a sufficient condition on119860(119911)guaranteeing that all solutions of (1) are in analytic Morreyspace is shown by using representation formula

It is also known that BMOA space is a subspace of Blochspace and then the properties of Bloch space were used in theproof of [12 Theorem 3] However it was proved that analyticMorrey space and Bloch space are different in [16]Thereforewe need different methods from Section 6 of [12] to deal withthe case of analytic Morrey space the following results areproved

eorem 6 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatsup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (31)

is sufficiently small and

sup119911isinD

|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2 lt 3 minus 1205822 (32)

Then all solutions of (1) are inL2120582(D)eorem 7 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatthe conditions of Theorem 6 are satisfied and

lim|119886|997888rarr1minus

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0 (33)

and

lim|119911|997888rarr1minus

(|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2) = 0 (34)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 6 and 7 we recall thedefinition of Carleson measure For a non-negative measure120583 on D it is called a Carleson measure if

10038171003817100381710038171205831003817100381710038171003817 š sup119868sub120597D

(120583 (119878 (119868))|119868| )12 lt infin (35)

Moreover if in addition

lim|119868|997888rarr0

120583 (119878 (119868))|119868| = 0 (36)

then 120583 is called a compact Carleson measureFor the proof of Theorems 6 and 7 the following lemmas

are needed

Lemma 8 (sse [14]) Suppose that 120583 is a Carleson measureThen

10038171003817100381710038171205831003817100381710038171003817 asymp sup119886isinD

intD

100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889120583 (119911) (37)

Journal of Function Spaces 5

Lemma 9 (see [23]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) thenintD

1003816100381610038161003816119891 (119911)1003816100381610038161003816119901 (1 minus |119911|2)119902 119889119898 (119911)≲ (1003816100381610038161003816119891 (0)1003816100381610038161003816119901 + int

D

100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119901+119902 119889119898 (119911)) (38)

Lemma 10 (see [24]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) then10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+2≲ int

D

10038161003816100381610038161003816119891(119899) (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119902 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (39)

Lemma 11 (see [22]) Suppose that 0 lt 119901 119902 lt infin Then for119891 isin A(D)sup119886isinD

10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902 (40)

and

sup119886isinD

10038161003816100381610038161003816119891(119899+1) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+119901 (41)

are comparable

Lemma 12 Let 0 lt 120582 lt 1 and 119891 isin L2120582 Then

100381710038171003817100381711989110038171003817100381710038172L2120582 ≲ (sup119886isinD

(100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)3minus120582)+ sup

119886isinD((1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119886 (119911)210038161003816100381610038161003816) 119889119898 (119911))) (42)

Proof An auxiliary function is constructed as follows

119892 (119911) = 1198911015840 (120590119886 (119911)) (1 minus |119886|2)(1 minus 119886119911)2 (43)

By Lemmas 1 and 9100381710038171003817100381711989110038171003817100381710038172L2120582 asymp (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 ∘ 120590119886 (119911)10038161003816100381610038162)sdot 100381610038161003816100381610038161205901015840119886 (119911)100381610038161003816100381610038162 119889119898 (119911) = (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2 119889119898 (119911)1 minus |119911|2 = (1minus |119886|2)1minus120582 int

D

1003816100381610038161003816119892 (119911)10038161003816100381610038162 (1 minus |119911|2) 119889119898 (119911) ≲ (1003816100381610038161003816119892 (0)10038161003816100381610038162sdot (1 minus |119886|2)1minus120582 + (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911))

(44)

where |119892(0)|2 = |1198911015840(119886)|2(1 minus |119886|2)2

Note that

(1 minus |119886|2)1minus120582 intD

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911) le 1198681 + 1198682 (45)

where

1198681 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161003816(1198911015840 (120590119886 (119911)))10158401003816100381610038161003816100381610038162|1 minus 119886119911|4 (1 minus |119886|2)2 (1 minus |119911|2)3 119889119898 (119911)(46)

and

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus |119886|2)2|1 minus 119886119911|6 (1 minus |119911|2)3 119889119898 (119911) (47)

Clearly

1198681 ≲ (1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)41 minus |119911|2 119889119898 (119911)

= (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(48)

It follows from Lemmas 10 and 11 that

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2|1 minus 119886119911|2 (1 minus |119911|2) 119889119898 (119911)le (1 minus |119886|2)1minus120582 sup

119886isinD100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)2

sdot intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911) ≲ (1 minus |119886|2)1minus120582

sdot sup119886isinD

1003816100381610038161003816100381611989110158401015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)4intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)≲ (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)sdot int

D

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)

(49)

It is easy to get that sup119886isinD intD((1 minus |119911|2)|1 minus 119886119911|2)119889119898(119911) isfinite Then the lemma is completely proved

6 Journal of Function Spaces

Lemma 13 (see [25]) Suppose that 120583 is a positive Borelmeasure and 0 lt 119901 lt infin Then 120583 is a Carleson measure if andonly if 119867119901(D) sub 119871119901(119889120583) Namely if there exists a continuousincluding mapping 119894 119867119901(D) 997888rarr 119871119901(119889120583) then

10038171003817100381710038171198911003817100381710038171003817119871119901(119889120583) ≲ 10038171003817100381710038171205831003817100381710038171003817 10038171003817100381710038171198911003817100381710038171003817119867119901(D) (50)

Lemma 14 (see [18]) Let 0 lt 120582 lt 1 If 119891 isin L2120582 then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817L2120582

(1 minus |119911|2)(1minus120582)2 119911 isin D (51)

Proof of Theorem 6 We can obtain that all solutions 119891 of (1)belong toB(3minus120582)2 by [11 Corollary 4] and (32) It follows fromLemma 12 that

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 ≲ sup119886isinD

((1 minus |119886|2)3minus120582 100381610038161003816100381610038161198911015840 (120588119911)100381610038161003816100381610038162 1205882+ (1 minus |119886|2)1minus120582sdot int

D

1205884 1003816100381610038161003816100381611989110158401015840 (120588119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911))≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + sup

119886isinD(1 minus |119886|2)1minus120582 sdot int

D

10038161003816100381610038161003816119891120588 (119911) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)+ sup

119886isinD(1 minus |119886|2)1minus120582 10038161003816100381610038161003816119891120588 (119886)100381610038161003816100381610038162 int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + 1198691+ 1198692

(52)

where 119891120588(119911) = 119891(120588119911) By Lemmas 1 8 and 13

1198691 ≲ sup119886isinD

(1 minus |119886|2)1minus120582 intD

10038161003816100381610038161003816119891120588 (120590 (119911)) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 sdot (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889119898 (119911)≲ sup

119886isinD(1 minus |119886|2)1minus120582 (10038171003817100381710038171003817119891120588 (120590 (119911)) minus 119891120588 (119886)1003817100381710038171003817100381721198672

sdot sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119887

∘ 120590minus1119886 (119911)100381610038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119888isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119888 (119911)10038161003816100381610038162) 119889119898 (119911)

(53)

where 119889120583(119911) = |119860(120588120590119886(119911))|2(1 minus |120590119886(119911)|2)3|1205901015840119886(119911)|119889119898(119911) is aCarleson measure

It follows from Lemma 14 that

1198692 ≲ sup119886isinD

(1 minus |119886|2)1minus12058210038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

(1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(54)

Denote

119869 (119886 120588) = intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) 0 lt 120588 lt 1

(55)

For |119886| le 12 the estimate is trivial Let 12 le |119886| le 1(2minus120588)Since |1 minus 119886119911| le 2|1 minus 119886119911120588| for |119911| le 120588119869 (119886 120588)

= intD(0120588)

|119860 (119911)|2 (1 minus 10038161003816100381610038161003816100381610038161003816 119911120588100381610038161003816100381610038161003816100381610038162)2 1 minus |119886|210038161003816100381610038161 minus 119886 (119911120588)1003816100381610038161003816

119889119898 (119911)1205882le 41205882 intD |119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)le 16int

D|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(56)

for any 12 lt 120588 lt 1 Let 1(2 minus 120588) le |119886| le 1 Now119869 (119886 120588) le sup

119886isinD|119860 (119911)|2 (1 minus |119911|2)4

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(57)

Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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4 Journal of Function Spaces

where 119862 is a positive constant and 0 lt 120588 lt 1 It follows fromFubinirsquos theorem and the reproducing formula that

sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

= sup119911isinD

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0(int

D

120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) 1198701015840120577(120578)119889119898 (120578))119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816)

≳ sup119911isinD

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(26)

Therefore we deduce 119891 isin B2(1+120572) by the above formula and(25) and letting 120588 997888rarr 1minusProof ofTheorem 4 ByTheorem 3 we know that119891 isin B2(1+120572)Fubinirsquos theorem and the reproducing formula yield

lim|119911|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119911|)2(1+120572)sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816)

≳ lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

(27)

The condition of Theorem 4 implies

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 = 0 (28)

By using the similar reason as in the proof of Theorem 3 wehave

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 ≲ 1003817100381710038171003817100381711989112058810038171003817100381710038171003817B2(1+120572) (1 minus |119911|)2(1+120572)

sdot 10038161003816100381610038161003816100381610038161003816intD 120596120574120572 (10038161003816100381610038161205781003816100381610038161003816) (int119911

01198701015840

120577(120578)119860 (120588120577) 119889120577) 119889119898 (120578)10038161003816100381610038161003816100381610038161003816

+ 10038161003816100381610038161003816119891120588 (0)10038161003816100381610038161003816 (1 minus |119911|)2(1+120572) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816 + (1 minus |119911|)2(1+120572)

sdot 100381610038161003816100381610038161198911015840120588 (0)10038161003816100381610038161003816 119911 isin D

(29)

Then by the condition of Theorem 4

lim|119911|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119911|)2(1+120572) 100381610038161003816100381610038161198911015840120588 (119911)10038161003816100381610038161003816 = 0 (30)

Therefore the assertion follows

4 Sufficient Conditions ofSolutions in L2120582(D) (L2120582

0 (D))In the section some sufficient conditions on 119860(119911) whichguarantee all solutions of (1) belong to analytic Morrey space

are obtained by using two ideas On the one hand a sufficientcondition on119860(119911) guaranteeing that all solutions of (1) belongto BMOA is obtained in [12 Theorem 3] here we study thecondition on 119860(119911) which guarantee that all solutions of (1)are in analytic Morrey space by using similar idea in [12Theorem3]On the other hand a sufficient condition on119860(119911)guaranteeing that all solutions of (1) are in analytic Morreyspace is shown by using representation formula

It is also known that BMOA space is a subspace of Blochspace and then the properties of Bloch space were used in theproof of [12 Theorem 3] However it was proved that analyticMorrey space and Bloch space are different in [16]Thereforewe need different methods from Section 6 of [12] to deal withthe case of analytic Morrey space the following results areproved

eorem 6 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatsup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (31)

is sufficiently small and

sup119911isinD

|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2 lt 3 minus 1205822 (32)

Then all solutions of (1) are inL2120582(D)eorem 7 Let 119860(119911) isin A(D) and 0 lt 120582 lt 1 Suppose thatthe conditions of Theorem 6 are satisfied and

lim|119886|997888rarr1minus

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0 (33)

and

lim|119911|997888rarr1minus

(|119860 (119911)| (1 minus |119911|)(5minus120582)2 int|119911|

0

119889119903(1 minus 119903)(3minus120582)2) = 0 (34)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 6 and 7 we recall thedefinition of Carleson measure For a non-negative measure120583 on D it is called a Carleson measure if

10038171003817100381710038171205831003817100381710038171003817 š sup119868sub120597D

(120583 (119878 (119868))|119868| )12 lt infin (35)

Moreover if in addition

lim|119868|997888rarr0

120583 (119878 (119868))|119868| = 0 (36)

then 120583 is called a compact Carleson measureFor the proof of Theorems 6 and 7 the following lemmas

are needed

Lemma 8 (sse [14]) Suppose that 120583 is a Carleson measureThen

10038171003817100381710038171205831003817100381710038171003817 asymp sup119886isinD

intD

100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889120583 (119911) (37)

Journal of Function Spaces 5

Lemma 9 (see [23]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) thenintD

1003816100381610038161003816119891 (119911)1003816100381610038161003816119901 (1 minus |119911|2)119902 119889119898 (119911)≲ (1003816100381610038161003816119891 (0)1003816100381610038161003816119901 + int

D

100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119901+119902 119889119898 (119911)) (38)

Lemma 10 (see [24]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) then10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+2≲ int

D

10038161003816100381610038161003816119891(119899) (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119902 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (39)

Lemma 11 (see [22]) Suppose that 0 lt 119901 119902 lt infin Then for119891 isin A(D)sup119886isinD

10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902 (40)

and

sup119886isinD

10038161003816100381610038161003816119891(119899+1) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+119901 (41)

are comparable

Lemma 12 Let 0 lt 120582 lt 1 and 119891 isin L2120582 Then

100381710038171003817100381711989110038171003817100381710038172L2120582 ≲ (sup119886isinD

(100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)3minus120582)+ sup

119886isinD((1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119886 (119911)210038161003816100381610038161003816) 119889119898 (119911))) (42)

Proof An auxiliary function is constructed as follows

119892 (119911) = 1198911015840 (120590119886 (119911)) (1 minus |119886|2)(1 minus 119886119911)2 (43)

By Lemmas 1 and 9100381710038171003817100381711989110038171003817100381710038172L2120582 asymp (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 ∘ 120590119886 (119911)10038161003816100381610038162)sdot 100381610038161003816100381610038161205901015840119886 (119911)100381610038161003816100381610038162 119889119898 (119911) = (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2 119889119898 (119911)1 minus |119911|2 = (1minus |119886|2)1minus120582 int

D

1003816100381610038161003816119892 (119911)10038161003816100381610038162 (1 minus |119911|2) 119889119898 (119911) ≲ (1003816100381610038161003816119892 (0)10038161003816100381610038162sdot (1 minus |119886|2)1minus120582 + (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911))

(44)

where |119892(0)|2 = |1198911015840(119886)|2(1 minus |119886|2)2

Note that

(1 minus |119886|2)1minus120582 intD

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911) le 1198681 + 1198682 (45)

where

1198681 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161003816(1198911015840 (120590119886 (119911)))10158401003816100381610038161003816100381610038162|1 minus 119886119911|4 (1 minus |119886|2)2 (1 minus |119911|2)3 119889119898 (119911)(46)

and

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus |119886|2)2|1 minus 119886119911|6 (1 minus |119911|2)3 119889119898 (119911) (47)

Clearly

1198681 ≲ (1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)41 minus |119911|2 119889119898 (119911)

= (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(48)

It follows from Lemmas 10 and 11 that

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2|1 minus 119886119911|2 (1 minus |119911|2) 119889119898 (119911)le (1 minus |119886|2)1minus120582 sup

119886isinD100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)2

sdot intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911) ≲ (1 minus |119886|2)1minus120582

sdot sup119886isinD

1003816100381610038161003816100381611989110158401015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)4intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)≲ (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)sdot int

D

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)

(49)

It is easy to get that sup119886isinD intD((1 minus |119911|2)|1 minus 119886119911|2)119889119898(119911) isfinite Then the lemma is completely proved

6 Journal of Function Spaces

Lemma 13 (see [25]) Suppose that 120583 is a positive Borelmeasure and 0 lt 119901 lt infin Then 120583 is a Carleson measure if andonly if 119867119901(D) sub 119871119901(119889120583) Namely if there exists a continuousincluding mapping 119894 119867119901(D) 997888rarr 119871119901(119889120583) then

10038171003817100381710038171198911003817100381710038171003817119871119901(119889120583) ≲ 10038171003817100381710038171205831003817100381710038171003817 10038171003817100381710038171198911003817100381710038171003817119867119901(D) (50)

Lemma 14 (see [18]) Let 0 lt 120582 lt 1 If 119891 isin L2120582 then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817L2120582

(1 minus |119911|2)(1minus120582)2 119911 isin D (51)

Proof of Theorem 6 We can obtain that all solutions 119891 of (1)belong toB(3minus120582)2 by [11 Corollary 4] and (32) It follows fromLemma 12 that

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 ≲ sup119886isinD

((1 minus |119886|2)3minus120582 100381610038161003816100381610038161198911015840 (120588119911)100381610038161003816100381610038162 1205882+ (1 minus |119886|2)1minus120582sdot int

D

1205884 1003816100381610038161003816100381611989110158401015840 (120588119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911))≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + sup

119886isinD(1 minus |119886|2)1minus120582 sdot int

D

10038161003816100381610038161003816119891120588 (119911) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)+ sup

119886isinD(1 minus |119886|2)1minus120582 10038161003816100381610038161003816119891120588 (119886)100381610038161003816100381610038162 int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + 1198691+ 1198692

(52)

where 119891120588(119911) = 119891(120588119911) By Lemmas 1 8 and 13

1198691 ≲ sup119886isinD

(1 minus |119886|2)1minus120582 intD

10038161003816100381610038161003816119891120588 (120590 (119911)) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 sdot (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889119898 (119911)≲ sup

119886isinD(1 minus |119886|2)1minus120582 (10038171003817100381710038171003817119891120588 (120590 (119911)) minus 119891120588 (119886)1003817100381710038171003817100381721198672

sdot sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119887

∘ 120590minus1119886 (119911)100381610038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119888isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119888 (119911)10038161003816100381610038162) 119889119898 (119911)

(53)

where 119889120583(119911) = |119860(120588120590119886(119911))|2(1 minus |120590119886(119911)|2)3|1205901015840119886(119911)|119889119898(119911) is aCarleson measure

It follows from Lemma 14 that

1198692 ≲ sup119886isinD

(1 minus |119886|2)1minus12058210038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

(1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(54)

Denote

119869 (119886 120588) = intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) 0 lt 120588 lt 1

(55)

For |119886| le 12 the estimate is trivial Let 12 le |119886| le 1(2minus120588)Since |1 minus 119886119911| le 2|1 minus 119886119911120588| for |119911| le 120588119869 (119886 120588)

= intD(0120588)

|119860 (119911)|2 (1 minus 10038161003816100381610038161003816100381610038161003816 119911120588100381610038161003816100381610038161003816100381610038162)2 1 minus |119886|210038161003816100381610038161 minus 119886 (119911120588)1003816100381610038161003816

119889119898 (119911)1205882le 41205882 intD |119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)le 16int

D|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(56)

for any 12 lt 120588 lt 1 Let 1(2 minus 120588) le |119886| le 1 Now119869 (119886 120588) le sup

119886isinD|119860 (119911)|2 (1 minus |119911|2)4

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(57)

Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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Journal of Function Spaces 5

Lemma 9 (see [23]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) thenintD

1003816100381610038161003816119891 (119911)1003816100381610038161003816119901 (1 minus |119911|2)119902 119889119898 (119911)≲ (1003816100381610038161003816119891 (0)1003816100381610038161003816119901 + int

D

100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119901+119902 119889119898 (119911)) (38)

Lemma 10 (see [24]) Let 0 lt 119901 119902 lt infin If 119891 isin A(D) then10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+2≲ int

D

10038161003816100381610038161003816119891(119899) (119911)10038161003816100381610038161003816119901 (1 minus |119911|2)119902 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (39)

Lemma 11 (see [22]) Suppose that 0 lt 119901 119902 lt infin Then for119891 isin A(D)sup119886isinD

10038161003816100381610038161003816119891(119899) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902 (40)

and

sup119886isinD

10038161003816100381610038161003816119891(119899+1) (119886)10038161003816100381610038161003816119901 (1 minus |119886|2)119902+119901 (41)

are comparable

Lemma 12 Let 0 lt 120582 lt 1 and 119891 isin L2120582 Then

100381710038171003817100381711989110038171003817100381710038172L2120582 ≲ (sup119886isinD

(100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)3minus120582)+ sup

119886isinD((1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119886 (119911)210038161003816100381610038161003816) 119889119898 (119911))) (42)

Proof An auxiliary function is constructed as follows

119892 (119911) = 1198911015840 (120590119886 (119911)) (1 minus |119886|2)(1 minus 119886119911)2 (43)

By Lemmas 1 and 9100381710038171003817100381711989110038171003817100381710038172L2120582 asymp (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= (1 minus |119886|2)1minus120582 int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 ∘ 120590119886 (119911)10038161003816100381610038162)sdot 100381610038161003816100381610038161205901015840119886 (119911)100381610038161003816100381610038162 119889119898 (119911) = (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2 119889119898 (119911)1 minus |119911|2 = (1minus |119886|2)1minus120582 int

D

1003816100381610038161003816119892 (119911)10038161003816100381610038162 (1 minus |119911|2) 119889119898 (119911) ≲ (1003816100381610038161003816119892 (0)10038161003816100381610038162sdot (1 minus |119886|2)1minus120582 + (1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911))

(44)

where |119892(0)|2 = |1198911015840(119886)|2(1 minus |119886|2)2

Note that

(1 minus |119886|2)1minus120582 intD

100381610038161003816100381610038161198921015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)3 119889119898 (119911) le 1198681 + 1198682 (45)

where

1198681 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161003816(1198911015840 (120590119886 (119911)))10158401003816100381610038161003816100381610038162|1 minus 119886119911|4 (1 minus |119886|2)2 (1 minus |119911|2)3 119889119898 (119911)(46)

and

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus |119886|2)2|1 minus 119886119911|6 (1 minus |119911|2)3 119889119898 (119911) (47)

Clearly

1198681 ≲ (1 minus |119886|2)1minus120582

sdot intD

1003816100381610038161003816100381611989110158401015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)41 minus |119911|2 119889119898 (119911)

= (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(48)

It follows from Lemmas 10 and 11 that

1198682 = (1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840 (120590119886 (119911))100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)2|1 minus 119886119911|2 (1 minus |119911|2) 119889119898 (119911)le (1 minus |119886|2)1minus120582 sup

119886isinD100381610038161003816100381610038161198911015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)2

sdot intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911) ≲ (1 minus |119886|2)1minus120582

sdot sup119886isinD

1003816100381610038161003816100381611989110158401015840 (119886)100381610038161003816100381610038162 (1 minus |119886|2)4intD

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)≲ (1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816100381611989110158401015840 (119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)sdot int

D

1 minus |119911|2|1 minus 119886119911|2 119889119898 (119911)

(49)

It is easy to get that sup119886isinD intD((1 minus |119911|2)|1 minus 119886119911|2)119889119898(119911) isfinite Then the lemma is completely proved

6 Journal of Function Spaces

Lemma 13 (see [25]) Suppose that 120583 is a positive Borelmeasure and 0 lt 119901 lt infin Then 120583 is a Carleson measure if andonly if 119867119901(D) sub 119871119901(119889120583) Namely if there exists a continuousincluding mapping 119894 119867119901(D) 997888rarr 119871119901(119889120583) then

10038171003817100381710038171198911003817100381710038171003817119871119901(119889120583) ≲ 10038171003817100381710038171205831003817100381710038171003817 10038171003817100381710038171198911003817100381710038171003817119867119901(D) (50)

Lemma 14 (see [18]) Let 0 lt 120582 lt 1 If 119891 isin L2120582 then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817L2120582

(1 minus |119911|2)(1minus120582)2 119911 isin D (51)

Proof of Theorem 6 We can obtain that all solutions 119891 of (1)belong toB(3minus120582)2 by [11 Corollary 4] and (32) It follows fromLemma 12 that

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 ≲ sup119886isinD

((1 minus |119886|2)3minus120582 100381610038161003816100381610038161198911015840 (120588119911)100381610038161003816100381610038162 1205882+ (1 minus |119886|2)1minus120582sdot int

D

1205884 1003816100381610038161003816100381611989110158401015840 (120588119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911))≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + sup

119886isinD(1 minus |119886|2)1minus120582 sdot int

D

10038161003816100381610038161003816119891120588 (119911) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)+ sup

119886isinD(1 minus |119886|2)1minus120582 10038161003816100381610038161003816119891120588 (119886)100381610038161003816100381610038162 int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + 1198691+ 1198692

(52)

where 119891120588(119911) = 119891(120588119911) By Lemmas 1 8 and 13

1198691 ≲ sup119886isinD

(1 minus |119886|2)1minus120582 intD

10038161003816100381610038161003816119891120588 (120590 (119911)) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 sdot (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889119898 (119911)≲ sup

119886isinD(1 minus |119886|2)1minus120582 (10038171003817100381710038171003817119891120588 (120590 (119911)) minus 119891120588 (119886)1003817100381710038171003817100381721198672

sdot sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119887

∘ 120590minus1119886 (119911)100381610038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119888isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119888 (119911)10038161003816100381610038162) 119889119898 (119911)

(53)

where 119889120583(119911) = |119860(120588120590119886(119911))|2(1 minus |120590119886(119911)|2)3|1205901015840119886(119911)|119889119898(119911) is aCarleson measure

It follows from Lemma 14 that

1198692 ≲ sup119886isinD

(1 minus |119886|2)1minus12058210038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

(1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(54)

Denote

119869 (119886 120588) = intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) 0 lt 120588 lt 1

(55)

For |119886| le 12 the estimate is trivial Let 12 le |119886| le 1(2minus120588)Since |1 minus 119886119911| le 2|1 minus 119886119911120588| for |119911| le 120588119869 (119886 120588)

= intD(0120588)

|119860 (119911)|2 (1 minus 10038161003816100381610038161003816100381610038161003816 119911120588100381610038161003816100381610038161003816100381610038162)2 1 minus |119886|210038161003816100381610038161 minus 119886 (119911120588)1003816100381610038161003816

119889119898 (119911)1205882le 41205882 intD |119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)le 16int

D|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(56)

for any 12 lt 120588 lt 1 Let 1(2 minus 120588) le |119886| le 1 Now119869 (119886 120588) le sup

119886isinD|119860 (119911)|2 (1 minus |119911|2)4

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(57)

Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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6 Journal of Function Spaces

Lemma 13 (see [25]) Suppose that 120583 is a positive Borelmeasure and 0 lt 119901 lt infin Then 120583 is a Carleson measure if andonly if 119867119901(D) sub 119871119901(119889120583) Namely if there exists a continuousincluding mapping 119894 119867119901(D) 997888rarr 119871119901(119889120583) then

10038171003817100381710038171198911003817100381710038171003817119871119901(119889120583) ≲ 10038171003817100381710038171205831003817100381710038171003817 10038171003817100381710038171198911003817100381710038171003817119867119901(D) (50)

Lemma 14 (see [18]) Let 0 lt 120582 lt 1 If 119891 isin L2120582 then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817L2120582

(1 minus |119911|2)(1minus120582)2 119911 isin D (51)

Proof of Theorem 6 We can obtain that all solutions 119891 of (1)belong toB(3minus120582)2 by [11 Corollary 4] and (32) It follows fromLemma 12 that

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 ≲ sup119886isinD

((1 minus |119886|2)3minus120582 100381610038161003816100381610038161198911015840 (120588119911)100381610038161003816100381610038162 1205882+ (1 minus |119886|2)1minus120582sdot int

D

1205884 1003816100381610038161003816100381611989110158401015840 (120588119911)100381610038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911))≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + sup

119886isinD(1 minus |119886|2)1minus120582 sdot int

D

10038161003816100381610038161003816119891120588 (119911) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)+ sup

119886isinD(1 minus |119886|2)1minus120582 10038161003816100381610038161003816119891120588 (119886)100381610038161003816100381610038162 int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172B(3minus120582)2 + 1198691+ 1198692

(52)

where 119891120588(119911) = 119891(120588119911) By Lemmas 1 8 and 13

1198691 ≲ sup119886isinD

(1 minus |119886|2)1minus120582 intD

10038161003816100381610038161003816119891120588 (120590 (119911)) minus 119891120588 (119886)100381610038161003816100381610038162sdot 1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 sdot (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816 119889119898 (119911)≲ sup

119886isinD(1 minus |119886|2)1minus120582 (10038171003817100381710038171003817119891120588 (120590 (119911)) minus 119891120588 (119886)1003817100381710038171003817100381721198672

sdot sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588120590119886 (119911))10038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)3 100381610038161003816100381610038161205901015840119886 (119911)10038161003816100381610038161003816sdot 100381610038161003816100381610038161205901015840119887 (119911)10038161003816100381610038161003816 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

(sup119887isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 10038161003816100381610038161003816120590119887

∘ 120590minus1119886 (119911)100381610038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119888isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1minus 1003816100381610038161003816120590119888 (119911)10038161003816100381610038162) 119889119898 (119911)

(53)

where 119889120583(119911) = |119860(120588120590119886(119911))|2(1 minus |120590119886(119911)|2)3|1205901015840119886(119911)|119889119898(119911) is aCarleson measure

It follows from Lemma 14 that

1198692 ≲ sup119886isinD

(1 minus |119886|2)1minus12058210038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

(1 minus |119886|2)1minus120582sdot int

D

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)= 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582sdot sup119886isinD

intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(54)

Denote

119869 (119886 120588) = intD

1003816100381610038161003816119860 (120588119911)10038161003816100381610038162 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) 0 lt 120588 lt 1

(55)

For |119886| le 12 the estimate is trivial Let 12 le |119886| le 1(2minus120588)Since |1 minus 119886119911| le 2|1 minus 119886119911120588| for |119911| le 120588119869 (119886 120588)

= intD(0120588)

|119860 (119911)|2 (1 minus 10038161003816100381610038161003816100381610038161003816 119911120588100381610038161003816100381610038161003816100381610038162)2 1 minus |119886|210038161003816100381610038161 minus 119886 (119911120588)1003816100381610038161003816

119889119898 (119911)1205882le 41205882 intD |119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)le 16int

D|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(56)

for any 12 lt 120588 lt 1 Let 1(2 minus 120588) le |119886| le 1 Now119869 (119886 120588) le sup

119886isinD|119860 (119911)|2 (1 minus |119911|2)4

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(57)

Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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Journal of Function Spaces 7

By Lemma 10 we get

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot intD

(1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(1 minus 100381610038161003816100381612058811991110038161003816100381610038162)4 119889119898 (119911)

(58)

By [12 Theorem 3]

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus 119904)3 (1 minus |119886|)(1 minus 120588119904)4 (1 minus |119886| 119904)119889119904

≲ sup119886isinD

intD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot int1

0

(1 minus |119886|)(1 minus 120588119904) (1 minus |119886| 119904)119889119904= sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

sdot 1 minus |119886||119886| minus 120588 log(1 + |119886| minus 1205881 minus |119886| )

(59)

Since 120588 le 2minus1|119886| ((1minus|119886|)(|119886|minus120588)) log(1+(|119886|minus120588)(1minus|119886|))is bounded Therefore

119869 (119886 120588)≲ sup

119886isinDintD|119860 (119911)|2 (1 minus |119911|2)2 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) (60)

If (31) is sufficiently small then the norm 1198911205882L2120582 is uni-formly bounded for 12 lt 120588 lt 1 By letting 120588 997888rarr 1minus wereduce 119891 isin L2120582

Proof ofTheorem 7 We can get that all solutions of (1) belongto B

(3minus120582)20 by the similar reason as in the proof of [11

Theorem 1] here it is omitted The next proof is also similarto the proof of Theorem 6 here it is also omitted

Next we consider the estimate of coefficient similarly toSection 8 in [12] in which the well-known representationformula

ℎ (119911) = 12120587 int2120587

0

ℎ (119890119894119904)1 minus 119911119890minus119894119904 119889119904 119911 isin D (61)

is used where ℎ isin 1198671(D) it is found in [26 Theorem 36]Here we prove the following results

eorem 15 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose thatlim

120588997888rarr1minussup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)(62)

is bounded and

lim120588997888rarr1minus

sup119886isinD

intD (1 minus1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

(63)

is sufficiently small where ℎ120588119911(119904) = int119911

0(119860(120588120589)(1 minus 119904120589))119889120589

Then all solutions of (1) are inL2120582(D)eorem 16 Let 119860 isin A(D) and 0 lt 120582 le 1 Suppose that theconditions of Theorem 15 are satisfied and

lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) = 0lim

|119886|997888rarr1minuslim

120588997888rarr1minusintD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

= 0

(64)

Then all solutions of (1) are inL21205820 (D)

In order to prove Theorems 15 and 16 the followinglemma is needed

Lemma 17 (see [12]) If 119891 119892 isin 1198672(D) then12120587 int

2120587

0119891 (119890119894119905) 119892 (119890119894119905)119889119905

= 2intD

1198911015840 (119904) 1198921015840 (119904) log 1|119904|119889119898 (119904) + 119891 (0) 119892 (0)(65)

Proof of Theorem 15 Let 119891 be any solution of (1) Then 119891120588is analytic in D and satisfies 11989110158401015840

120588 (119911) + 1205882119860(120588119911)119891120588(119911) = 0Therefore

1198911015840120588 (119911) = minusint119911

01205882119891120588 (120589) 119860 (120588120589) 119889120589 + 1198911015840

120588 (0) 119911 isin D (66)

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

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Submit your manuscripts atwwwhindawicom

8 Journal of Function Spaces

By the representation formula and Fubinirsquos theorem

1198911015840120588 (119911) = minus 12120587 int

2120587

0119891120588 (119890119894119904)int119911

0

1205882119860 (120588120589)1 minus 119890minus119894119904120589 119889120589119889119904 + 1198911015840120588 (0)

= minus 12058822120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119904 + 1198911015840

120588 (0) 119911 isin D(67)

where

ℎ120588119911 (119904) = int119911

0

119860 (120588120589)1 minus 119904120589 119889120589 (68)

Since 119891120588 ℎ120588119911 isin 1198672(D) Lemma 17 implies

12120587 int2120587

0119891120588 (119890119894119904) ℎ120588119911 (119890119894119904)119889119905

= 2intD

1198911015840120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904) + 119891120588 (0) ℎ120588119911 (0)

(69)

So

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 le 8 10038161003816100381610038161003816100381610038161003816intD 1198911015840

120588 (119904) ℎ1015840120588119911 (119904) log 1|119904|119889119898 (119904)100381610038161003816100381610038161003816100381610038162

+ 2 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0) minus 1198911015840120588 (0)100381610038161003816100381610038162

≲ intD

100381610038161003816100381610038161198911015840120588 (119904)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162) 119889119898 (119904)

sdot intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904)

+ 4 10038161003816100381610038161003816119891120588 (0) ℎ120588119911 (0)100381610038161003816100381610038162 + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(70)

Therefore by Lemma 1

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 asymp sup119886isinD

((1 minus |119886|2)1minus120582

sdot intD

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816ℎ120588119911 (0)100381610038161003816100381610038162 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162sdot sup119886isinD

(1 minus |119886|2)1minus120582 intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)

(71)

Then

10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582 (1 minus 119862 sup119886isinD

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)(intD

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162

sdot (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911))≲ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 sup

119886isinD(1 minus |119886|2)1minus120582 int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot 10038161003816100381610038161003816100381610038161003816int

119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911) + 4 100381610038161003816100381610038161198911015840120588 (0)100381610038161003816100381610038162

(72)

where 119862 is a positive constant By letting 120588 997888rarr 1minus (62) and(63) the assertion follows

Proof of Theorem 16 By the assumption and Theorem 15 weget 119891 isin L2120582 By using the similar reason as the proof ofTheorem 15 and the condition of Theorem 16 we have

lim|119886|997888rarr1minus

lim120588997888rarr1minus

((1 minus |119886|2)1minus120582sdot int

D

100381610038161003816100381610038161198911015840120588 (119911)100381610038161003816100381610038162 (1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911)) ≲ 10038171003817100381710038171003817119891120588100381710038171003817100381710038172L2120582

sdot lim|119886|997888rarr1minus

lim120588997888rarr1minus

intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162)sdot (int

D

10038161003816100381610038161003816ℎ1015840120588119911 (119904)100381610038161003816100381610038162 (log |119904|)2(1 minus 1003816100381610038161003816120590119886 (119904)10038161003816100381610038162)119889119898 (119904))119889119898 (119911)

+ 4 10038161003816100381610038161003816119891120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

lim120588997888rarr1minus

(1 minus |119886|2)1minus120582

sdot intD

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 10038161003816100381610038161003816100381610038161003816int119911

0119860 (120588120577) 11988912057710038161003816100381610038161003816100381610038161003816

2 119889119898 (119911)+ 4 100381610038161003816100381610038161198911015840

120588 (0)100381610038161003816100381610038162 lim|119886|997888rarr1minus

(1 minus |119886|2)1minus120582sdot int

D

(1 minus 1003816100381610038161003816120590119886 (119911)10038161003816100381610038162) 119889119898 (119911) = 0

(73)

The assertion follows by Lemmas 1 and 2

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

All authors have contributed equally to this manuscript Allauthors read and approved the final manuscript

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Function Spaces 9

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11861023 and 11501142)the Foundation of Doctoral Research Program of GuizhouNormal University 2016 the Foundation of Qian CengCi Innovative Talents of Guizhou Province 2016 and theFoundation of Science and Technology of Guizhou Provinceof China (Grant no [2015]2112)References

[1] D Benbourenane and L Sons ldquoOn global solutions of complexdifferential equations in the unit diskrdquo Complex VariablesTheory and Application vol 49 no 49 pp 913ndash925 2004

[2] T Cao and H Yi ldquoThe growth of solutions of linear differentialequations with coefficients of iterated order in the unit discrdquoJournal of Mathematical Analysis and Applications vol 319 no1 pp 278ndash294 2006

[3] Z-X Chen and K H Shon ldquoThe growth of solutions ofdifferential equations with coefficients of small growth in thediscrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 285ndash304 2004

[4] I Chyzhykov G G Gundersen and J Heittokangas ldquoLineardifferential equations and logarithmic derivative estimatesrdquoProceedings of the London Mathematical Society Third Seriesvol 86 no 3 pp 735ndash754 2003

[5] J Heittokangas R Korhonen and J Rattya ldquoFast growingsolutions of linear differential equations in the unit discrdquoResultsin Mathematics vol 49 no 3-4 pp 265ndash278 2006

[6] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with coefficients in weighted Bergman and Hardyspacesrdquo Transactions of the AmericanMathematical Society vol360 no 2 pp 1035ndash1055 2008

[7] J Heittokangas ldquoOn complex differential equations in the unitdiscrdquo Annales Academiae Scientiarum Fennicae Series A IMathematica Dissertationes vol 122 pp 1ndash54 2000

[8] J Grohn ldquoNew applications of Picardrsquos successive approxima-tionsrdquo Bulletin des Sciences Mathematiques vol 135 no 5 pp475ndash487 2011

[9] C Pommerenke ldquoOn the mean growth of the solutions ofcomplex linear differential equations in the diskrdquo ComplexVariables Theory and Application vol 1 pp 23ndash38 1982

[10] J Heittokangas R Korhonen and J Rattya ldquoLinear differentialequations with solutions in the dirichlet type subspace of thehardy spacerdquo Nagoya Mathematical Journal vol 187 pp 91ndash1132007

[11] J-M Huusko T Korhonen and A Reijonen ldquoLinear differen-tial equations with solutions in the growth space119867infin

120596 rdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 41 no 1 pp399ndash416 2016

[12] J Grohn J-M Huusko and J Rattya ldquoLinear differentialequations with slowly growing solutionsrdquo Transactions of theAmerican Mathematical Society vol 370 no 10 pp 7201ndash72272018

[13] J Pelaez ldquoSmall weighted Bergman spacesrdquo in Proceedingsof the Summer School in Complex and Harmonic Analysisand Related Topics Publications of the University of EasternFinland Reports and Studies in Forestry and Natural Sciences2016

[14] J Liu and Z Lou ldquoCarleson measure for analytic Morreyspacesrdquo Nonlinear Analysis vol 125 pp 423ndash432 2015

[15] Z Wu and C Xie ldquoQ spaces and Morrey spacesrdquo Journal ofFunctional Analysis vol 201 no 1 pp 282ndash297 2003

[16] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[17] C B Morrey ldquoOn the solutions of quasi-linear elliptic partialdifferential equationsrdquo Transactions of the AmericanMathemat-ical Society vol 43 no 1 pp 126ndash166 1938

[18] P Li J Liu and Z Lou ldquoIntegral operators on analytic Morreyspacesrdquo Science ChinaMathematics vol 57 no 9 pp 1961ndash19742014

[19] H Wulan and J Zhou ldquo119876119870 and Morrey type spacesrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 38 no 1 pp193ndash207 2013

[20] J Xiao Geometric 119876119901 Functions Frontiers in MathematicsBirkhaauser Basel Switzerland 2006

[21] J Pau and J Pelaez ldquoEmbedding theorems and integrationoperators on Bergman spaces with rapidly decreasing weightsrdquoJournal of Functional Analysis vol 259 no 10 pp 2727ndash27562010

[22] K H Zhu ldquoBloch type spaces of analytic functionsrdquo RockyMountain Journal of Mathematics vol 23 no 3 pp 1143ndash11771993

[23] J Pau and J Pelaez ldquoVolterra type operators on Bergman spaceswith exponential weightsrdquoContemporaryMathematics vol 561pp 239ndash252 2012

[24] HWulan and J Zhou ldquoThe higher order derivatives of119876119870 typespacesrdquo Journal of Mathematical Analysis and Applications vol332 no 2 pp 1216ndash1228 2007

[25] D Girela ldquoAnalytic functions of bounded mean oscillationrdquoin Complex functions spaces Report Series No 4 pp 61ndash71University of Joensuu Department of Mathematics 2001

[26] P DurenTheory of Hp Spaces Academic Press New York NYUSA 1970

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom