Research ArticleGrowth Analysis of Composite Entire and MeromorphicFunctions in the Light of Their Relative Orders

Sanjib Kumar Datta1 Tanmay Biswas2 and Chinmay Biswas3

1 Department of Mathematics University of Kalyani Kalyani Nadia District West Bengal 741235 India2 Rajbari Rabindra Pally R N Tagore Road Krishnagar Nadia District West Bengal 741101 India3 Taraknagar Jamuna Sundari High School Taraknagar Hanskhali Nadia District West Bengal 741502 India

Correspondence should be addressed to Sanjib Kumar Datta sanjib kr dattayahoocoin

Received 29 June 2014 Accepted 20 August 2014 Published 29 October 2014

Academic Editor George L Karakostas

Copyright copy 2014 Sanjib Kumar Datta et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study some comparative growth properties of composite entire and meromorphic functions on the basis of their relative orders(relative lower orders)

1 Introduction

Let 119891 be meromorphic and 119892 be an entire function defined inthe open complex plane C The maximummodulus functioncorresponding to entire 119892 is defined as119872119892(119903) = max|119892(119911)| |119911| = 119903 For meromorphic 119891 119872119891(119903) cannot be defined as119891 is not analytic In this situation one may define anotherfunction119879119891(119903) known as Nevanlinnarsquos characteristic functionof 119891 playing the same role as maximummodulus function inthe following manner

119879119891 (119903) = 119873119891 (119903) + 119898119891 (119903) (1)

where the function 119873119891(119903 119886)(119873119891(119903 119886)) known as countingfunction of 119886-points (distinct 119886-points) of meromorphic 119891 isdefined as

119873119891 (119903 119886) = int119903

0

119899119891 (119905 119886) minus 119899119891 (0 119886)

119905119889119905 + 119899119891 (0 119886) log 119903

(119873119891 (119903 119886) = int119903

0

119899119891 (119905 119886) minus 119899119891 (0 119886)

119905119889119905 + 119899119891 (0 119886) log 119903)

(2)

Moreover we denote by 119899119891(119903 119886)(119899119891(119903 119886)) the number of 119886-points (distinct 119886-points) of 119891 in |119911| le 119903 and an infin-point isa pole of 119891 In many occasions 119873119891(119903infin) and 119873119891(119903infin) aredenoted by119873119891(119903) and119873119891(119903) respectively

The function 119898119891(119903infin) alternatively denoted by 119898119891(119903)known as the proximity function of 119891 is defined as follows

119898119891 (119903) =1

2120587int2120587

0

log+ 10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816 119889120579

where log+119909 = max (log119909 0) forall119909 ⩾ 0

(3)

Also we may denote119898(119903 1(119891 minus 119886)) by119898119891(119903 119886)When119891 is an entire function the Nevanlinnarsquos character-

istic function 119879119891(119903) of 119891 is defined as

119879119891 (119903) = 119898119891 (119903) (4)

Further if 119891 is a nonconstant entire function then119872119891(119903) and 119879119891(119903) are both strictly increasing and continuousfunctions of 119903 Also their inverses 119872minus1

119891(119903) (|119891(0)|infin) rarr

(0infin) and 119879minus1119891

(119879119891(0)infin) rarr (0infin) exist respectivelyand are such that lim119904rarrinfin119872

minus1

119892(119904) = infin and lim119904rarrinfin119879

minus1

119891(119904) =

infinIn this connection we just recall the following definition

which is relevant

Definition 1 (see [1]) A nonconstant entire function 119891 is saidto have the Property (A) if for any 120590 gt 1 and for all suffi-ciently large 119903 [119872119891(119903)]

2 le 119872119891(119903120590) holds For examples of

functions with or without the Property (A) one may see [1]

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 538327 6 pageshttpdxdoiorg1011552014538327

2 International Scholarly Research Notices

However for any two entire functions 119891 and 119892 the ratio119872119891(119903)119872119892(119903) as 119903 rarr infin is called the growth of 119891 withrespect to 119892 in terms of their maximum moduli Similarlywhen 119891 and 119892 are both meromorphic functions the ratio119879119891(119903)119879119892(119903) as 119903 rarr infin is called the growth of 119891 withrespect to 119892 in terms of their Nevanlinnarsquos characteristicfunctions The notion of the growth indicators such as orderand lower order of entire ormeromorphic functionswhich aregenerally used in computational purpose is defined in termsof their growth with respect to the exponential function as thefollowing

Definition 2 The order 120588119891 (the lower order 120582119891) of an entirefunction 119891 is defined as

120588119891 = lim sup119903rarrinfin

log log119872119891 (119903)log log119872exp 119911 (119903)

= lim sup119903rarrinfin

log[2]119872119891 (119903)log 119903

(120582119891 = lim inf119903rarrinfin

log log119872119891 (119903)log log119872exp 119911 (119903)

= lim inf119903rarrinfin

log[2]119872119891 (119903)log 119903

)

(5)

where log[119896]119909 = log(log[119896minus1]119909) for 119896 = 1 2 3 and log[0]119909 =119909 Further if 119891 is a meromorphic function one can easilyverify that

120588119891 = lim sup119903rarrinfin

log119879119891 (119903)log119879exp 119911 (119903)

= lim sup119903rarrinfin

log119879119891 (119903)log (119903120587)

= lim sup119903rarrinfin

log119879119891 (119903)log 119903 + 119874 (1)

(120582119891 = lim inf119903rarrinfin

log119879119891 (119903)log119879exp 119911 (119903)

= lim inf119903rarrinfin

log119879119891 (119903)log (119903120587)

= lim inf119903rarrinfin

log119879119891 (119903)log 119903 + 119874 (1)

)

(6)

Bernal [1 2] introduced the definition of relative order ofan entire function119891with respect to another entire function119892denoted by 120588119892(119891) to avoid comparing growth just with exp 119911as follows

120588119892 (119891)

= inf 120583 gt 0 119872119891 (119903) lt 119872119892 (119903120583) forall119903 gt 1199030 (120583) gt 0

= lim sup119903rarrinfin

log119872minus1119892119872119891 (119903)

log 119903

(7)

Thedefinition coincideswith the classical one [3] if119892(119911) =exp 119911

Similarly one can define the relative lower order of anentire function 119891 with respect to another entire function 119892denoted by 120582119892(119891) as follows

120582119892 (119891) = lim inf119903rarrinfin

log119872minus1119892119872119891 (119903)

log 119903 (8)

Extending this notion Lahiri andBanerjee [4] introducedthe definition of relative order of a meromorphic functionwith respect to an entire function in the following way

Definition 3 (see [4]) Let 119891 be any meromorphic functionand 119892 be any entire function The relative order of 119891 withrespect to 119892 is defined as

120588119892 (119891)

= inf 120583 gt 0 119879119891 (119903) lt 119879119892 (119903120583) for all sufficiently large 119903

= lim sup119903rarrinfin

log119879minus1119892119879119891 (119903)

log 119903

(9)

Likewise one can define the relative lower order of a mero-morphic function 119891 with respect to an entire function 119892denoted by 120582119892(119891) as follows

120582119892 (119891) = lim inf119903rarrinfin

log119879minus1119892119879119891 (119903)

log 119903 (10)

It is known (cf [4]) that if 119892(119911) = exp 119911 then Definition 3coincides with the classical definition of the order of ameromorphic function 119891

For entire and meromorphic functions the notions oftheir growth indicators such as order and lower order areclassical in complex analysis and during the past decadesseveral researchers have already been exploring their studiesin the area of comparative growth properties of compositeentire and meromorphic functions in different directionsusing the classical growth indicators But at that time theconcepts of relative orders and relative lower orders of entireand meromorphic functions and their technical advantagesof not comparing with the growths of exp 119911 are not at allknown to the researchers of this areaTherefore the studies ofthe growths of composite entire and meromorphic functionsin the light of their relative orders and relative lower ordersare the prime concern of this paper In fact some light hasalready been thrown on such type of works by Datta et al [5]We do not explain the standard definitions and notations ofthe theory of entire and meromorphic functions as those areavailable in [6 7]

2 Lemmas

In this section we present some lemmas which will be neededin the sequel

Lemma 1 (see [8]) Let 119891 be meromorphic and 119892 be entirethen for all sufficiently large values of 119903

119879119891∘119892 (119903) ⩽ 1 + 119900 (1)119879119892 (119903)

log119872119892 (119903)119879119891 (119872119892 (119903)) (11)

International Scholarly Research Notices 3

Lemma 2 (see [9]) Let119891 bemeromorphic and 119892 be entire andsuppose that 0 lt 120583 lt 120588119892 le infin Then for a sequence of valuesof 119903 tending to infinity

119879119891∘119892 (119903) ge 119879119891 (exp (119903120583)) (12)

Lemma 3 (see [10]) Let 119891 be meromorphic and 119892 be entiresuch that 0 lt 120588119892 lt infin and 0 lt 120582119891 Then for a sequence ofvalues of 119903 tending to infinity

119879119891∘119892 (119903) gt 119879119892 (exp (119903120583)) (13)

where 0 lt 120583 lt 120588119892

Lemma 4 (see [11]) Let 119891 be an entire function which satisfiesthe Property (A) 120573 gt 0 120575 gt 1 and 120572 gt 2 Then

120573119879119891 (119903) lt 119879119891 (120572119903120575) (14)

3 Theorems

In this section we present the main results of the paper

Theorem 1 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin and let 119892 bean entire function with finite order If ℎ satisfies the Property(A) then for every positive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 where 120583 gt (1 + 120572) 120588119892

(15)

Proof Let us suppose that 120573 gt 2 and 120575 gt 1 If 1 + 120572 le 0 thenthe theorem is obvious We consider 1 + 120572 gt 0

Since 119879minus1ℎ(119903) is an increasing function of 119903 it follows from

Lemma 1 Lemma 4 and the inequality119879119892(119903) le log119872119892(119903) cf[6] for all sufficiently large values of 119903 that

119879minus1ℎ119879119891∘119892 (119903) ⩽ 119879minus1

ℎ[1 + 119900 (1) 119879119891 (119872119892 (119903))]

that is 119879minus1ℎ119879119891∘119892 (119903) ⩽ 120573[119879minus1

ℎ119879119891 (119872119892 (119903))]

120575

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 log119879minus1

ℎ119879119891 (119872119892 (119903)) + 119874 (1)

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874 (1)

(16)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119891 (exp 119903

120583) ge (120582ℎ (119891) minus 120576) 119903120583 (17)

Hence for all sufficiently large values of 119903 we obtain from (16)and (17) that

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

le[120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874(1)]1+120572

(120582ℎ (119891) minus 120576) 119903120583

(18)

where we choose 0 lt 120576 lt min120582ℎ(119891) (120583(1 + 120572)) minus 120588119892So from (18) we obtain that

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 (19)

This proves the theorem

Remark 2 InTheorem 1 if we take the condition 0 lt 120588ℎ(119891) ltinfin instead of 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin the theorem remainstrue with ldquolimit inferiorrdquo in place of ldquolimitrdquo

In viewofTheorem 1 the following theorem can be carriedout

Theorem 3 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions where 119892 is of finite order and 120582ℎ(119892) gt0 120588ℎ(119891) lt infin If ℎ satisfies the Property (A) then for everypositive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119892 (exp 119903120583)

= 0 119908ℎ119890119903119890 120583 gt (1 + 120572) 120588119892

(20)

The proof is omitted

Remark 4 In Theorem 3 if we take the condition 120588ℎ(119892) gt 0instead of 120582ℎ(119892) gt 0 the theorem remains true with ldquolimitrdquoreplaced by ldquolimit inferiorrdquo

Theorem 5 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt infin Also suppose that ℎ satisfies the Property(A) Then for a sequence of values of 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119891 (exp 119903

120583) (21)

Proof Let us consider 120575 gt 1 Since 119879minus1ℎ(119903) is an increasing

function of 119903 it follows from Lemma 1 that for a sequence ofvalues of 119903 tending to infinity

log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120582119892+120576 + 119874 (1) (22)

4 International Scholarly Research Notices

Now from (17) and (22) it follows for a sequence of values of119903 tending to infinity that

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

ge(120582ℎ (119891) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120582119892+120576 + 119874 (1)

(23)

As 120582119892 lt 120583 we can choose 120576 (gt0) in such a way that

120582119892 + 120576 lt 120583 lt 120588119892 (24)

Thus from (23) and (24) we obtain that

lim sup119903rarrinfin

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

= infin (25)

Now from (25) we obtain for a sequence of values of 119903 tendingto infinity and also for119870 gt 1

119879minus1ℎ119879119891 (exp 119903

120583) gt 119879minus1ℎ119879119891∘119892 (119903) (26)

Thus the theorem follows

In the line of Theorem 5 we may state the followingtheorem without its proof

Theorem 6 Let 119892 and ℎ be any two entire functions with120582ℎ(119892) gt 0 and let 119891 be a meromorphic function with finiterelative order with respect to ℎ Also suppose that 120582119892 lt 120583 lt infinand ℎ satisfies the Property (A) Then for a sequence of valuesof 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119892 (exp 119903

120583) (27)

As an application of Theorem 5 and Lemma 2 we maystate the following theorem

Theorem 7 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt 120588119892 If ℎ satisfies the Property (A) then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

(28)

The proof is omittedSimilary in view of Theorem 6 and Lemma 3 the

following theorem can be carried out

Theorem 8 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin 0 lt120582ℎ(119892) le 120588ℎ(119892) lt infin and 0 lt 120582119892 lt 120583 lt 120588119892 lt infin Moreover ℎsatisfies the Property (A) Then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

(29)

The proof is omitted

Theorem 9 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (30)

where 0 lt 120583 lt 120588119892

Proof Let 0 lt 120583 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing

function of 119903 it follows from Lemma 2 for a sequence ofvalues of 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge log119879minus1

ℎ119879119891 (exp (119903

120583

))

that is log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891) minus 120576) 119903

120583

(31)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119892 (exp 119903

120583) le (120588ℎ (119892) + 120576) 119903120583 (32)

So combining (31) and (32) we obtain for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

ge(120582ℎ (119891) minus 120576) 119903

120583

(120588ℎ (119892) + 120576) 119903120583 (33)

Since 120583 lt 120583 it follows from (33) that

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (34)

Hence the theorem follows

Corollary 10 Under the assumptions of Theorem 9

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (35)

Proof In view of Theorem 9 we get for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge 119870 log119879minus1

ℎ119879119892 (exp 119903

120583) for 119870 gt 1

that is 119879minus1ℎ119879119891∘119892 (119903) ge log 119879minus1

ℎ119879119892 (exp 119903

120583)119870

(36)

from which the corollary follows

Theorem 11 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120588ℎ(119892) lt infin (ii)120582ℎ(119891) gt 0 and (iii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119892 (exp 119903120583) sdot log119879minus1ℎ 119879119892 (119903)

= infin (37)

where 0 lt 120583 lt 120588119892

International Scholarly Research Notices 5

Proof From the definition of relative order and relative lowerorder we obtain for arbitrary positive 120576 and for all sufficientlylarge values of 119903 that

log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891 ∘ 119892) minus 120576) log 119903

log119879minus1ℎ119879119892 (119903) le (120588ℎ (119892) + 120576) log 119903

(38)

Therefore from (38) it follows for all sufficiently large valuesof 119903 that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge(120582ℎ (119891 ∘ 119892) minus 120576) log 119903(120588ℎ (119892) + 120576) log 119903

that is lim inf119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge120582ℎ (119891 ∘ 119892)

120588ℎ (119892)

(39)

Thus the theorem follows from (34) and (39)

Similarly one may state the following theorems andcorollary without their proofs as those can be carried out inthe line of Theorems 9 and 11 and Corollary 10 respectively

Theorem 12 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119891 (exp 119903120583)

= infin (40)

where 0 lt 120583 lt 120588119892

Theorem 13 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand (ii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119891 (exp 119903120583) sdot log119879minus1ℎ 119879119891 (119903)

= infin (41)

where 0 lt 120583 lt 120588119892

Corollary 14 Under the assumptions of Theorem 12

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (42)

Theorem 15 Let 119891 be a meromorphic function and let ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

= infin (43)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

Proof Let 0 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing function of

119903 it follows from (31) for a sequence of values of 119903 tending toinfinity that

log[2]119879minus1ℎ119879119891∘119892 (119903) ge 119874 (1) + 120583

log 119903 (44)

So for a sequence of values of 119903 tending to infinity we get fromabove that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861) ge 119874 (1) + 120583119903119861 (45)

Again we have for all sufficiently large values of 119903 that

log[2]119879minus1ℎ119879119891 (exp 119903

120583) le 119874 (1) + 120583 log 119903 (46)

Now combining (45) and (46) we obtain for a sequence ofvalues of 119903 tending to infinity that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

ge119874 (1) + 120583119903119861

119874 (1) + 120583 log 119903 (47)

from which the theorem follows

In view of Theorem 15 the following theorem can becarried out

Theorem 16 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119892 (exp 119903120583)

= infin (48)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

The proof is omitted

Theorem 17 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder such that 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin (49)

Proof Since 120588119892 lt 120588119896 we can choose 120576 (gt0) in such a way that

120588119892 + 120576 lt 120583 lt 120588119896 minus 120576 (50)

As 119879minus1119897(119903) is an increasing function of 119903 it follows from

Lemma 2 for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903) ge log119879minus1

119897119879ℎ (exp 119903

120583)

where 0 lt 120583 lt 120588119896 le infin

that is log119879minus1119897119879ℎ∘119896 (119903) ge (120582119897 (ℎ) minus 120576) 119903

120583

(51)

Now from the definition of relative order of 119891 with respect to119897 we have for arbitrary positive 120576 and for all sufficiently largevalues of 119903 that

log119879minus1119897119879119891 (119903) le (120588119897 (119891) + 120576) log 119903 (52)

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Scholarly Research Notices

However for any two entire functions 119891 and 119892 the ratio119872119891(119903)119872119892(119903) as 119903 rarr infin is called the growth of 119891 withrespect to 119892 in terms of their maximum moduli Similarlywhen 119891 and 119892 are both meromorphic functions the ratio119879119891(119903)119879119892(119903) as 119903 rarr infin is called the growth of 119891 withrespect to 119892 in terms of their Nevanlinnarsquos characteristicfunctions The notion of the growth indicators such as orderand lower order of entire ormeromorphic functionswhich aregenerally used in computational purpose is defined in termsof their growth with respect to the exponential function as thefollowing

Definition 2 The order 120588119891 (the lower order 120582119891) of an entirefunction 119891 is defined as

120588119891 = lim sup119903rarrinfin

log log119872119891 (119903)log log119872exp 119911 (119903)

= lim sup119903rarrinfin

log[2]119872119891 (119903)log 119903

(120582119891 = lim inf119903rarrinfin

log log119872119891 (119903)log log119872exp 119911 (119903)

= lim inf119903rarrinfin

log[2]119872119891 (119903)log 119903

)

(5)

where log[119896]119909 = log(log[119896minus1]119909) for 119896 = 1 2 3 and log[0]119909 =119909 Further if 119891 is a meromorphic function one can easilyverify that

120588119891 = lim sup119903rarrinfin

log119879119891 (119903)log119879exp 119911 (119903)

= lim sup119903rarrinfin

log119879119891 (119903)log (119903120587)

= lim sup119903rarrinfin

log119879119891 (119903)log 119903 + 119874 (1)

(120582119891 = lim inf119903rarrinfin

log119879119891 (119903)log119879exp 119911 (119903)

= lim inf119903rarrinfin

log119879119891 (119903)log (119903120587)

= lim inf119903rarrinfin

log119879119891 (119903)log 119903 + 119874 (1)

)

(6)

Bernal [1 2] introduced the definition of relative order ofan entire function119891with respect to another entire function119892denoted by 120588119892(119891) to avoid comparing growth just with exp 119911as follows

120588119892 (119891)

= inf 120583 gt 0 119872119891 (119903) lt 119872119892 (119903120583) forall119903 gt 1199030 (120583) gt 0

= lim sup119903rarrinfin

log119872minus1119892119872119891 (119903)

log 119903

(7)

Thedefinition coincideswith the classical one [3] if119892(119911) =exp 119911

Similarly one can define the relative lower order of anentire function 119891 with respect to another entire function 119892denoted by 120582119892(119891) as follows

120582119892 (119891) = lim inf119903rarrinfin

log119872minus1119892119872119891 (119903)

log 119903 (8)

Extending this notion Lahiri andBanerjee [4] introducedthe definition of relative order of a meromorphic functionwith respect to an entire function in the following way

Definition 3 (see [4]) Let 119891 be any meromorphic functionand 119892 be any entire function The relative order of 119891 withrespect to 119892 is defined as

120588119892 (119891)

= inf 120583 gt 0 119879119891 (119903) lt 119879119892 (119903120583) for all sufficiently large 119903

= lim sup119903rarrinfin

log119879minus1119892119879119891 (119903)

log 119903

(9)

Likewise one can define the relative lower order of a mero-morphic function 119891 with respect to an entire function 119892denoted by 120582119892(119891) as follows

120582119892 (119891) = lim inf119903rarrinfin

log119879minus1119892119879119891 (119903)

log 119903 (10)

It is known (cf [4]) that if 119892(119911) = exp 119911 then Definition 3coincides with the classical definition of the order of ameromorphic function 119891

For entire and meromorphic functions the notions oftheir growth indicators such as order and lower order areclassical in complex analysis and during the past decadesseveral researchers have already been exploring their studiesin the area of comparative growth properties of compositeentire and meromorphic functions in different directionsusing the classical growth indicators But at that time theconcepts of relative orders and relative lower orders of entireand meromorphic functions and their technical advantagesof not comparing with the growths of exp 119911 are not at allknown to the researchers of this areaTherefore the studies ofthe growths of composite entire and meromorphic functionsin the light of their relative orders and relative lower ordersare the prime concern of this paper In fact some light hasalready been thrown on such type of works by Datta et al [5]We do not explain the standard definitions and notations ofthe theory of entire and meromorphic functions as those areavailable in [6 7]

2 Lemmas

In this section we present some lemmas which will be neededin the sequel

Lemma 1 (see [8]) Let 119891 be meromorphic and 119892 be entirethen for all sufficiently large values of 119903

119879119891∘119892 (119903) ⩽ 1 + 119900 (1)119879119892 (119903)

log119872119892 (119903)119879119891 (119872119892 (119903)) (11)

International Scholarly Research Notices 3

Lemma 2 (see [9]) Let119891 bemeromorphic and 119892 be entire andsuppose that 0 lt 120583 lt 120588119892 le infin Then for a sequence of valuesof 119903 tending to infinity

119879119891∘119892 (119903) ge 119879119891 (exp (119903120583)) (12)

Lemma 3 (see [10]) Let 119891 be meromorphic and 119892 be entiresuch that 0 lt 120588119892 lt infin and 0 lt 120582119891 Then for a sequence ofvalues of 119903 tending to infinity

119879119891∘119892 (119903) gt 119879119892 (exp (119903120583)) (13)

where 0 lt 120583 lt 120588119892

Lemma 4 (see [11]) Let 119891 be an entire function which satisfiesthe Property (A) 120573 gt 0 120575 gt 1 and 120572 gt 2 Then

120573119879119891 (119903) lt 119879119891 (120572119903120575) (14)

3 Theorems

In this section we present the main results of the paper

Theorem 1 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin and let 119892 bean entire function with finite order If ℎ satisfies the Property(A) then for every positive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 where 120583 gt (1 + 120572) 120588119892

(15)

Proof Let us suppose that 120573 gt 2 and 120575 gt 1 If 1 + 120572 le 0 thenthe theorem is obvious We consider 1 + 120572 gt 0

Since 119879minus1ℎ(119903) is an increasing function of 119903 it follows from

Lemma 1 Lemma 4 and the inequality119879119892(119903) le log119872119892(119903) cf[6] for all sufficiently large values of 119903 that

119879minus1ℎ119879119891∘119892 (119903) ⩽ 119879minus1

ℎ[1 + 119900 (1) 119879119891 (119872119892 (119903))]

that is 119879minus1ℎ119879119891∘119892 (119903) ⩽ 120573[119879minus1

ℎ119879119891 (119872119892 (119903))]

120575

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 log119879minus1

ℎ119879119891 (119872119892 (119903)) + 119874 (1)

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874 (1)

(16)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119891 (exp 119903

120583) ge (120582ℎ (119891) minus 120576) 119903120583 (17)

Hence for all sufficiently large values of 119903 we obtain from (16)and (17) that

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

le[120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874(1)]1+120572

(120582ℎ (119891) minus 120576) 119903120583

(18)

where we choose 0 lt 120576 lt min120582ℎ(119891) (120583(1 + 120572)) minus 120588119892So from (18) we obtain that

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 (19)

This proves the theorem

Remark 2 InTheorem 1 if we take the condition 0 lt 120588ℎ(119891) ltinfin instead of 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin the theorem remainstrue with ldquolimit inferiorrdquo in place of ldquolimitrdquo

In viewofTheorem 1 the following theorem can be carriedout

Theorem 3 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions where 119892 is of finite order and 120582ℎ(119892) gt0 120588ℎ(119891) lt infin If ℎ satisfies the Property (A) then for everypositive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119892 (exp 119903120583)

= 0 119908ℎ119890119903119890 120583 gt (1 + 120572) 120588119892

(20)

The proof is omitted

Remark 4 In Theorem 3 if we take the condition 120588ℎ(119892) gt 0instead of 120582ℎ(119892) gt 0 the theorem remains true with ldquolimitrdquoreplaced by ldquolimit inferiorrdquo

Theorem 5 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt infin Also suppose that ℎ satisfies the Property(A) Then for a sequence of values of 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119891 (exp 119903

120583) (21)

Proof Let us consider 120575 gt 1 Since 119879minus1ℎ(119903) is an increasing

function of 119903 it follows from Lemma 1 that for a sequence ofvalues of 119903 tending to infinity

log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120582119892+120576 + 119874 (1) (22)

4 International Scholarly Research Notices

Now from (17) and (22) it follows for a sequence of values of119903 tending to infinity that

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

ge(120582ℎ (119891) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120582119892+120576 + 119874 (1)

(23)

As 120582119892 lt 120583 we can choose 120576 (gt0) in such a way that

120582119892 + 120576 lt 120583 lt 120588119892 (24)

Thus from (23) and (24) we obtain that

lim sup119903rarrinfin

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

= infin (25)

Now from (25) we obtain for a sequence of values of 119903 tendingto infinity and also for119870 gt 1

119879minus1ℎ119879119891 (exp 119903

120583) gt 119879minus1ℎ119879119891∘119892 (119903) (26)

Thus the theorem follows

In the line of Theorem 5 we may state the followingtheorem without its proof

Theorem 6 Let 119892 and ℎ be any two entire functions with120582ℎ(119892) gt 0 and let 119891 be a meromorphic function with finiterelative order with respect to ℎ Also suppose that 120582119892 lt 120583 lt infinand ℎ satisfies the Property (A) Then for a sequence of valuesof 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119892 (exp 119903

120583) (27)

As an application of Theorem 5 and Lemma 2 we maystate the following theorem

Theorem 7 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt 120588119892 If ℎ satisfies the Property (A) then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

(28)

The proof is omittedSimilary in view of Theorem 6 and Lemma 3 the

following theorem can be carried out

Theorem 8 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin 0 lt120582ℎ(119892) le 120588ℎ(119892) lt infin and 0 lt 120582119892 lt 120583 lt 120588119892 lt infin Moreover ℎsatisfies the Property (A) Then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

(29)

The proof is omitted

Theorem 9 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (30)

where 0 lt 120583 lt 120588119892

Proof Let 0 lt 120583 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing

function of 119903 it follows from Lemma 2 for a sequence ofvalues of 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge log119879minus1

ℎ119879119891 (exp (119903

120583

))

that is log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891) minus 120576) 119903

120583

(31)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119892 (exp 119903

120583) le (120588ℎ (119892) + 120576) 119903120583 (32)

So combining (31) and (32) we obtain for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

ge(120582ℎ (119891) minus 120576) 119903

120583

(120588ℎ (119892) + 120576) 119903120583 (33)

Since 120583 lt 120583 it follows from (33) that

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (34)

Hence the theorem follows

Corollary 10 Under the assumptions of Theorem 9

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (35)

Proof In view of Theorem 9 we get for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge 119870 log119879minus1

ℎ119879119892 (exp 119903

120583) for 119870 gt 1

that is 119879minus1ℎ119879119891∘119892 (119903) ge log 119879minus1

ℎ119879119892 (exp 119903

120583)119870

(36)

from which the corollary follows

Theorem 11 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120588ℎ(119892) lt infin (ii)120582ℎ(119891) gt 0 and (iii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119892 (exp 119903120583) sdot log119879minus1ℎ 119879119892 (119903)

= infin (37)

where 0 lt 120583 lt 120588119892

International Scholarly Research Notices 5

Proof From the definition of relative order and relative lowerorder we obtain for arbitrary positive 120576 and for all sufficientlylarge values of 119903 that

log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891 ∘ 119892) minus 120576) log 119903

log119879minus1ℎ119879119892 (119903) le (120588ℎ (119892) + 120576) log 119903

(38)

Therefore from (38) it follows for all sufficiently large valuesof 119903 that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge(120582ℎ (119891 ∘ 119892) minus 120576) log 119903(120588ℎ (119892) + 120576) log 119903

that is lim inf119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge120582ℎ (119891 ∘ 119892)

120588ℎ (119892)

(39)

Thus the theorem follows from (34) and (39)

Similarly one may state the following theorems andcorollary without their proofs as those can be carried out inthe line of Theorems 9 and 11 and Corollary 10 respectively

Theorem 12 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119891 (exp 119903120583)

= infin (40)

where 0 lt 120583 lt 120588119892

Theorem 13 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand (ii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119891 (exp 119903120583) sdot log119879minus1ℎ 119879119891 (119903)

= infin (41)

where 0 lt 120583 lt 120588119892

Corollary 14 Under the assumptions of Theorem 12

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (42)

Theorem 15 Let 119891 be a meromorphic function and let ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

= infin (43)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

Proof Let 0 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing function of

119903 it follows from (31) for a sequence of values of 119903 tending toinfinity that

log[2]119879minus1ℎ119879119891∘119892 (119903) ge 119874 (1) + 120583

log 119903 (44)

So for a sequence of values of 119903 tending to infinity we get fromabove that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861) ge 119874 (1) + 120583119903119861 (45)

Again we have for all sufficiently large values of 119903 that

log[2]119879minus1ℎ119879119891 (exp 119903

120583) le 119874 (1) + 120583 log 119903 (46)

Now combining (45) and (46) we obtain for a sequence ofvalues of 119903 tending to infinity that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

ge119874 (1) + 120583119903119861

119874 (1) + 120583 log 119903 (47)

from which the theorem follows

In view of Theorem 15 the following theorem can becarried out

Theorem 16 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119892 (exp 119903120583)

= infin (48)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

The proof is omitted

Theorem 17 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder such that 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin (49)

Proof Since 120588119892 lt 120588119896 we can choose 120576 (gt0) in such a way that

120588119892 + 120576 lt 120583 lt 120588119896 minus 120576 (50)

As 119879minus1119897(119903) is an increasing function of 119903 it follows from

Lemma 2 for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903) ge log119879minus1

119897119879ℎ (exp 119903

120583)

where 0 lt 120583 lt 120588119896 le infin

that is log119879minus1119897119879ℎ∘119896 (119903) ge (120582119897 (ℎ) minus 120576) 119903

120583

(51)

Now from the definition of relative order of 119891 with respect to119897 we have for arbitrary positive 120576 and for all sufficiently largevalues of 119903 that

log119879minus1119897119879119891 (119903) le (120588119897 (119891) + 120576) log 119903 (52)

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 3

Lemma 2 (see [9]) Let119891 bemeromorphic and 119892 be entire andsuppose that 0 lt 120583 lt 120588119892 le infin Then for a sequence of valuesof 119903 tending to infinity

119879119891∘119892 (119903) ge 119879119891 (exp (119903120583)) (12)

Lemma 3 (see [10]) Let 119891 be meromorphic and 119892 be entiresuch that 0 lt 120588119892 lt infin and 0 lt 120582119891 Then for a sequence ofvalues of 119903 tending to infinity

119879119891∘119892 (119903) gt 119879119892 (exp (119903120583)) (13)

where 0 lt 120583 lt 120588119892

Lemma 4 (see [11]) Let 119891 be an entire function which satisfiesthe Property (A) 120573 gt 0 120575 gt 1 and 120572 gt 2 Then

120573119879119891 (119903) lt 119879119891 (120572119903120575) (14)

3 Theorems

In this section we present the main results of the paper

Theorem 1 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin and let 119892 bean entire function with finite order If ℎ satisfies the Property(A) then for every positive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 where 120583 gt (1 + 120572) 120588119892

(15)

Proof Let us suppose that 120573 gt 2 and 120575 gt 1 If 1 + 120572 le 0 thenthe theorem is obvious We consider 1 + 120572 gt 0

Since 119879minus1ℎ(119903) is an increasing function of 119903 it follows from

Lemma 1 Lemma 4 and the inequality119879119892(119903) le log119872119892(119903) cf[6] for all sufficiently large values of 119903 that

119879minus1ℎ119879119891∘119892 (119903) ⩽ 119879minus1

ℎ[1 + 119900 (1) 119879119891 (119872119892 (119903))]

that is 119879minus1ℎ119879119891∘119892 (119903) ⩽ 120573[119879minus1

ℎ119879119891 (119872119892 (119903))]

120575

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 log119879minus1

ℎ119879119891 (119872119892 (119903)) + 119874 (1)

that is log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874 (1)

(16)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119891 (exp 119903

120583) ge (120582ℎ (119891) minus 120576) 119903120583 (17)

Hence for all sufficiently large values of 119903 we obtain from (16)and (17) that

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

le[120575 (120588ℎ (119891) + 120576) 119903

120588119892+120576 + 119874(1)]1+120572

(120582ℎ (119891) minus 120576) 119903120583

(18)

where we choose 0 lt 120576 lt min120582ℎ(119891) (120583(1 + 120572)) minus 120588119892So from (18) we obtain that

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119891 (exp 119903120583)

= 0 (19)

This proves the theorem

Remark 2 InTheorem 1 if we take the condition 0 lt 120588ℎ(119891) ltinfin instead of 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin the theorem remainstrue with ldquolimit inferiorrdquo in place of ldquolimitrdquo

In viewofTheorem 1 the following theorem can be carriedout

Theorem 3 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions where 119892 is of finite order and 120582ℎ(119892) gt0 120588ℎ(119891) lt infin If ℎ satisfies the Property (A) then for everypositive constant 120583 and each 120572 isin (minusinfininfin)

lim119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

1+120572

log119879minus1ℎ119879119892 (exp 119903120583)

= 0 119908ℎ119890119903119890 120583 gt (1 + 120572) 120588119892

(20)

The proof is omitted

Remark 4 In Theorem 3 if we take the condition 120588ℎ(119892) gt 0instead of 120582ℎ(119892) gt 0 the theorem remains true with ldquolimitrdquoreplaced by ldquolimit inferiorrdquo

Theorem 5 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt infin Also suppose that ℎ satisfies the Property(A) Then for a sequence of values of 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119891 (exp 119903

120583) (21)

Proof Let us consider 120575 gt 1 Since 119879minus1ℎ(119903) is an increasing

function of 119903 it follows from Lemma 1 that for a sequence ofvalues of 119903 tending to infinity

log119879minus1ℎ119879119891∘119892 (119903) ⩽ 120575 (120588ℎ (119891) + 120576) 119903

120582119892+120576 + 119874 (1) (22)

4 International Scholarly Research Notices

Now from (17) and (22) it follows for a sequence of values of119903 tending to infinity that

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

ge(120582ℎ (119891) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120582119892+120576 + 119874 (1)

(23)

As 120582119892 lt 120583 we can choose 120576 (gt0) in such a way that

120582119892 + 120576 lt 120583 lt 120588119892 (24)

Thus from (23) and (24) we obtain that

lim sup119903rarrinfin

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

= infin (25)

Now from (25) we obtain for a sequence of values of 119903 tendingto infinity and also for119870 gt 1

119879minus1ℎ119879119891 (exp 119903

120583) gt 119879minus1ℎ119879119891∘119892 (119903) (26)

Thus the theorem follows

In the line of Theorem 5 we may state the followingtheorem without its proof

Theorem 6 Let 119892 and ℎ be any two entire functions with120582ℎ(119892) gt 0 and let 119891 be a meromorphic function with finiterelative order with respect to ℎ Also suppose that 120582119892 lt 120583 lt infinand ℎ satisfies the Property (A) Then for a sequence of valuesof 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119892 (exp 119903

120583) (27)

As an application of Theorem 5 and Lemma 2 we maystate the following theorem

Theorem 7 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt 120588119892 If ℎ satisfies the Property (A) then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

(28)

The proof is omittedSimilary in view of Theorem 6 and Lemma 3 the

following theorem can be carried out

Theorem 8 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin 0 lt120582ℎ(119892) le 120588ℎ(119892) lt infin and 0 lt 120582119892 lt 120583 lt 120588119892 lt infin Moreover ℎsatisfies the Property (A) Then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

(29)

The proof is omitted

Theorem 9 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (30)

where 0 lt 120583 lt 120588119892

Proof Let 0 lt 120583 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing

function of 119903 it follows from Lemma 2 for a sequence ofvalues of 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge log119879minus1

ℎ119879119891 (exp (119903

120583

))

that is log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891) minus 120576) 119903

120583

(31)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119892 (exp 119903

120583) le (120588ℎ (119892) + 120576) 119903120583 (32)

So combining (31) and (32) we obtain for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

ge(120582ℎ (119891) minus 120576) 119903

120583

(120588ℎ (119892) + 120576) 119903120583 (33)

Since 120583 lt 120583 it follows from (33) that

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (34)

Hence the theorem follows

Corollary 10 Under the assumptions of Theorem 9

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (35)

Proof In view of Theorem 9 we get for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge 119870 log119879minus1

ℎ119879119892 (exp 119903

120583) for 119870 gt 1

that is 119879minus1ℎ119879119891∘119892 (119903) ge log 119879minus1

ℎ119879119892 (exp 119903

120583)119870

(36)

from which the corollary follows

Theorem 11 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120588ℎ(119892) lt infin (ii)120582ℎ(119891) gt 0 and (iii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119892 (exp 119903120583) sdot log119879minus1ℎ 119879119892 (119903)

= infin (37)

where 0 lt 120583 lt 120588119892

International Scholarly Research Notices 5

Proof From the definition of relative order and relative lowerorder we obtain for arbitrary positive 120576 and for all sufficientlylarge values of 119903 that

log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891 ∘ 119892) minus 120576) log 119903

log119879minus1ℎ119879119892 (119903) le (120588ℎ (119892) + 120576) log 119903

(38)

Therefore from (38) it follows for all sufficiently large valuesof 119903 that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge(120582ℎ (119891 ∘ 119892) minus 120576) log 119903(120588ℎ (119892) + 120576) log 119903

that is lim inf119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge120582ℎ (119891 ∘ 119892)

120588ℎ (119892)

(39)

Thus the theorem follows from (34) and (39)

Similarly one may state the following theorems andcorollary without their proofs as those can be carried out inthe line of Theorems 9 and 11 and Corollary 10 respectively

Theorem 12 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119891 (exp 119903120583)

= infin (40)

where 0 lt 120583 lt 120588119892

Theorem 13 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand (ii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119891 (exp 119903120583) sdot log119879minus1ℎ 119879119891 (119903)

= infin (41)

where 0 lt 120583 lt 120588119892

Corollary 14 Under the assumptions of Theorem 12

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (42)

Theorem 15 Let 119891 be a meromorphic function and let ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

= infin (43)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

Proof Let 0 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing function of

119903 it follows from (31) for a sequence of values of 119903 tending toinfinity that

log[2]119879minus1ℎ119879119891∘119892 (119903) ge 119874 (1) + 120583

log 119903 (44)

So for a sequence of values of 119903 tending to infinity we get fromabove that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861) ge 119874 (1) + 120583119903119861 (45)

Again we have for all sufficiently large values of 119903 that

log[2]119879minus1ℎ119879119891 (exp 119903

120583) le 119874 (1) + 120583 log 119903 (46)

Now combining (45) and (46) we obtain for a sequence ofvalues of 119903 tending to infinity that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

ge119874 (1) + 120583119903119861

119874 (1) + 120583 log 119903 (47)

from which the theorem follows

In view of Theorem 15 the following theorem can becarried out

Theorem 16 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119892 (exp 119903120583)

= infin (48)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

The proof is omitted

Theorem 17 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder such that 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin (49)

Proof Since 120588119892 lt 120588119896 we can choose 120576 (gt0) in such a way that

120588119892 + 120576 lt 120583 lt 120588119896 minus 120576 (50)

As 119879minus1119897(119903) is an increasing function of 119903 it follows from

Lemma 2 for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903) ge log119879minus1

119897119879ℎ (exp 119903

120583)

where 0 lt 120583 lt 120588119896 le infin

that is log119879minus1119897119879ℎ∘119896 (119903) ge (120582119897 (ℎ) minus 120576) 119903

120583

(51)

Now from the definition of relative order of 119891 with respect to119897 we have for arbitrary positive 120576 and for all sufficiently largevalues of 119903 that

log119879minus1119897119879119891 (119903) le (120588119897 (119891) + 120576) log 119903 (52)

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Scholarly Research Notices

Now from (17) and (22) it follows for a sequence of values of119903 tending to infinity that

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

ge(120582ℎ (119891) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120582119892+120576 + 119874 (1)

(23)

As 120582119892 lt 120583 we can choose 120576 (gt0) in such a way that

120582119892 + 120576 lt 120583 lt 120588119892 (24)

Thus from (23) and (24) we obtain that

lim sup119903rarrinfin

log119879minus1ℎ119879119891 (exp 119903

120583)

log119879minus1ℎ119879119891∘119892 (119903)

= infin (25)

Now from (25) we obtain for a sequence of values of 119903 tendingto infinity and also for119870 gt 1

119879minus1ℎ119879119891 (exp 119903

120583) gt 119879minus1ℎ119879119891∘119892 (119903) (26)

Thus the theorem follows

In the line of Theorem 5 we may state the followingtheorem without its proof

Theorem 6 Let 119892 and ℎ be any two entire functions with120582ℎ(119892) gt 0 and let 119891 be a meromorphic function with finiterelative order with respect to ℎ Also suppose that 120582119892 lt 120583 lt infinand ℎ satisfies the Property (A) Then for a sequence of valuesof 119903 tending to infinity

119879minus1ℎ119879119891∘119892 (119903) lt 119879minus1

ℎ119879119892 (exp 119903

120583) (27)

As an application of Theorem 5 and Lemma 2 we maystate the following theorem

Theorem 7 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions such that 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand 120582119892 lt 120583 lt 120588119892 If ℎ satisfies the Property (A) then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

(28)

The proof is omittedSimilary in view of Theorem 6 and Lemma 3 the

following theorem can be carried out

Theorem 8 Let 119891 be a meromorphic function and let 119892 ℎ beany two entire functions with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin 0 lt120582ℎ(119892) le 120588ℎ(119892) lt infin and 0 lt 120582119892 lt 120583 lt 120588119892 lt infin Moreover ℎsatisfies the Property (A) Then

lim inf119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

le 1 le lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

(29)

The proof is omitted

Theorem 9 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (30)

where 0 lt 120583 lt 120588119892

Proof Let 0 lt 120583 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing

function of 119903 it follows from Lemma 2 for a sequence ofvalues of 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge log119879minus1

ℎ119879119891 (exp (119903

120583

))

that is log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891) minus 120576) 119903

120583

(31)

Again for all sufficiently large values of 119903 we get that

log119879minus1ℎ119879119892 (exp 119903

120583) le (120588ℎ (119892) + 120576) 119903120583 (32)

So combining (31) and (32) we obtain for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

ge(120582ℎ (119891) minus 120576) 119903

120583

(120588ℎ (119892) + 120576) 119903120583 (33)

Since 120583 lt 120583 it follows from (33) that

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (exp 119903120583)

= infin (34)

Hence the theorem follows

Corollary 10 Under the assumptions of Theorem 9

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119892 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (35)

Proof In view of Theorem 9 we get for a sequence of valuesof 119903 tending to infinity that

log119879minus1ℎ119879119891∘119892 (119903) ge 119870 log119879minus1

ℎ119879119892 (exp 119903

120583) for 119870 gt 1

that is 119879minus1ℎ119879119891∘119892 (119903) ge log 119879minus1

ℎ119879119892 (exp 119903

120583)119870

(36)

from which the corollary follows

Theorem 11 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120588ℎ(119892) lt infin (ii)120582ℎ(119891) gt 0 and (iii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119892 (exp 119903120583) sdot log119879minus1ℎ 119879119892 (119903)

= infin (37)

where 0 lt 120583 lt 120588119892

International Scholarly Research Notices 5

Proof From the definition of relative order and relative lowerorder we obtain for arbitrary positive 120576 and for all sufficientlylarge values of 119903 that

log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891 ∘ 119892) minus 120576) log 119903

log119879minus1ℎ119879119892 (119903) le (120588ℎ (119892) + 120576) log 119903

(38)

Therefore from (38) it follows for all sufficiently large valuesof 119903 that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge(120582ℎ (119891 ∘ 119892) minus 120576) log 119903(120588ℎ (119892) + 120576) log 119903

that is lim inf119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge120582ℎ (119891 ∘ 119892)

120588ℎ (119892)

(39)

Thus the theorem follows from (34) and (39)

Similarly one may state the following theorems andcorollary without their proofs as those can be carried out inthe line of Theorems 9 and 11 and Corollary 10 respectively

Theorem 12 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119891 (exp 119903120583)

= infin (40)

where 0 lt 120583 lt 120588119892

Theorem 13 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand (ii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119891 (exp 119903120583) sdot log119879minus1ℎ 119879119891 (119903)

= infin (41)

where 0 lt 120583 lt 120588119892

Corollary 14 Under the assumptions of Theorem 12

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (42)

Theorem 15 Let 119891 be a meromorphic function and let ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

= infin (43)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

Proof Let 0 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing function of

119903 it follows from (31) for a sequence of values of 119903 tending toinfinity that

log[2]119879minus1ℎ119879119891∘119892 (119903) ge 119874 (1) + 120583

log 119903 (44)

So for a sequence of values of 119903 tending to infinity we get fromabove that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861) ge 119874 (1) + 120583119903119861 (45)

Again we have for all sufficiently large values of 119903 that

log[2]119879minus1ℎ119879119891 (exp 119903

120583) le 119874 (1) + 120583 log 119903 (46)

Now combining (45) and (46) we obtain for a sequence ofvalues of 119903 tending to infinity that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

ge119874 (1) + 120583119903119861

119874 (1) + 120583 log 119903 (47)

from which the theorem follows

In view of Theorem 15 the following theorem can becarried out

Theorem 16 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119892 (exp 119903120583)

= infin (48)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

The proof is omitted

Theorem 17 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder such that 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin (49)

Proof Since 120588119892 lt 120588119896 we can choose 120576 (gt0) in such a way that

120588119892 + 120576 lt 120583 lt 120588119896 minus 120576 (50)

As 119879minus1119897(119903) is an increasing function of 119903 it follows from

Lemma 2 for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903) ge log119879minus1

119897119879ℎ (exp 119903

120583)

where 0 lt 120583 lt 120588119896 le infin

that is log119879minus1119897119879ℎ∘119896 (119903) ge (120582119897 (ℎ) minus 120576) 119903

120583

(51)

Now from the definition of relative order of 119891 with respect to119897 we have for arbitrary positive 120576 and for all sufficiently largevalues of 119903 that

log119879minus1119897119879119891 (119903) le (120588119897 (119891) + 120576) log 119903 (52)

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 5

Proof From the definition of relative order and relative lowerorder we obtain for arbitrary positive 120576 and for all sufficientlylarge values of 119903 that

log119879minus1ℎ119879119891∘119892 (119903) ge (120582ℎ (119891 ∘ 119892) minus 120576) log 119903

log119879minus1ℎ119879119892 (119903) le (120588ℎ (119892) + 120576) log 119903

(38)

Therefore from (38) it follows for all sufficiently large valuesof 119903 that

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge(120582ℎ (119891 ∘ 119892) minus 120576) log 119903(120588ℎ (119892) + 120576) log 119903

that is lim inf119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119892 (119903)

ge120582ℎ (119891 ∘ 119892)

120588ℎ (119892)

(39)

Thus the theorem follows from (34) and (39)

Similarly one may state the following theorems andcorollary without their proofs as those can be carried out inthe line of Theorems 9 and 11 and Corollary 10 respectively

Theorem 12 Let 119891 be a meromorphic function and ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log119879minus1ℎ119879119891∘119892 (119903)

log119879minus1ℎ119879119891 (exp 119903120583)

= infin (40)

where 0 lt 120583 lt 120588119892

Theorem 13 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions such that (i) 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infinand (ii) 120582ℎ(119891 ∘ 119892) gt 0 Then

lim sup119903rarrinfin

[log119879minus1ℎ119879119891∘119892 (119903)]

2

log119879minus1ℎ119879119891 (exp 119903120583) sdot log119879minus1ℎ 119879119891 (119903)

= infin (41)

where 0 lt 120583 lt 120588119892

Corollary 14 Under the assumptions of Theorem 12

lim sup119903rarrinfin

119879minus1ℎ119879119891∘119892 (119903)

119879minus1ℎ119879119891 (exp 119903120583)

= infin 0 lt 120583 lt 120588119892 (42)

Theorem 15 Let 119891 be a meromorphic function and let ℎ be anentire function with 0 lt 120582ℎ(119891) le 120588ℎ(119891) lt infin Then for anyentire function 119892

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

= infin (43)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

Proof Let 0 lt 120583 lt 120588119892 As 119879minus1

ℎ(119903) is an increasing function of

119903 it follows from (31) for a sequence of values of 119903 tending toinfinity that

log[2]119879minus1ℎ119879119891∘119892 (119903) ge 119874 (1) + 120583

log 119903 (44)

So for a sequence of values of 119903 tending to infinity we get fromabove that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861) ge 119874 (1) + 120583119903119861 (45)

Again we have for all sufficiently large values of 119903 that

log[2]119879minus1ℎ119879119891 (exp 119903

120583) le 119874 (1) + 120583 log 119903 (46)

Now combining (45) and (46) we obtain for a sequence ofvalues of 119903 tending to infinity that

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119891 (exp 119903120583)

ge119874 (1) + 120583119903119861

119874 (1) + 120583 log 119903 (47)

from which the theorem follows

In view of Theorem 15 the following theorem can becarried out

Theorem 16 Let 119891 be a meromorphic function and let ℎ 119892 beany two entire functions with 120582ℎ(119891) gt 0 and 0 lt 120588ℎ(119892) lt infinThen

lim sup119903rarrinfin

log[2]119879minus1ℎ119879119891∘119892 (exp 119903

119861)

log[2]119879minus1ℎ119879119892 (exp 119903120583)

= infin (48)

where 0 lt 120583 lt 120588119892 and 119861 gt 0

The proof is omitted

Theorem 17 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder such that 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin (49)

Proof Since 120588119892 lt 120588119896 we can choose 120576 (gt0) in such a way that

120588119892 + 120576 lt 120583 lt 120588119896 minus 120576 (50)

As 119879minus1119897(119903) is an increasing function of 119903 it follows from

Lemma 2 for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903) ge log119879minus1

119897119879ℎ (exp 119903

120583)

where 0 lt 120583 lt 120588119896 le infin

that is log119879minus1119897119879ℎ∘119896 (119903) ge (120582119897 (ℎ) minus 120576) 119903

120583

(51)

Now from the definition of relative order of 119891 with respect to119897 we have for arbitrary positive 120576 and for all sufficiently largevalues of 119903 that

log119879minus1119897119879119891 (119903) le (120588119897 (119891) + 120576) log 119903 (52)

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Scholarly Research Notices

Now for any 120575 gt 1 we get from (16) (51) (52) and in view of(50) for a sequence of values of 119903 tending to infinity that

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

ge(120582119897 (ℎ) minus 120576) 119903

120583

120575 (120588ℎ (119891) + 120576) 119903120588119892+120576 + (120588119897 (119891) + 120576) log 119903 + 119874 (1)

that islog119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119891 (119903)

= infin

(53)

which proves the theorem

In the line of Theorem 17 the following theorem can becarried out

Theorem 18 Let 119897 be an entire function satisfying the Property(A) and let ℎ be a meromorphic function such that 120582119897(ℎ) gt 0Also let 119892 and 119896 be any two entire functions with finite nonzeroorder and also 120588119892 lt 120588119896 Then for every meromorphic function119891 with 0 lt 120588119897(119891) lt infin

lim sup119903rarrinfin

log119879minus1119897119879ℎ∘119896 (119903)

log119879minus1119897119879119891∘119892 (119903) + log119879minus1

119897119879119892 (119903)

= infin (54)

4 Conclusion

Actually this paper deals with the extension of the works onthe growth properties of composite entire and meromorphicfunctions on the basis of their relative orders and relative lowerorders These theories can also be modified by the treatmentof the notions of generalized relative orders (generalizedrelative lower orders) and (119901 119902)th relative orders ((119901 119902)threlative lower orders) Moreover some extensions of the sametype may be done in the light of slowly changing functions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Bernal ldquoOrden relative de crecimiento de funciones enterasrdquoCollectanea Mathematica vol 39 no 3 pp 209ndash229 1988

[2] L Bernal Crecimiento relativo de funciones enteras Contribu-cion al estudio de lasfunciones enteras con ndice exponencialfinito [Doctoral thesis] University of Seville Sevilla Spain1984

[3] E C Titchmarsh The Theory of Functions Oxford UniversityPress Oxford UK 2nd edition 1939

[4] B K Lahiri and D Banerjee ldquoRelative order of entire andmeromorphic functionsrdquo Proceedings of the National Academyof Sciences India A vol 69 no 3 pp 339ndash354 1999

[5] S K Datta T Biswas and S Bhattacharyya ldquoGrowth rates ofmeromorphic functions focusing relative orderrdquo Chinese Jour-nal of Mathematics vol 2014 Article ID 582082 7 pages 2014

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs The Clarendon Press Oxford UK 1964

[7] G Valiron Lectures on the General Theory of Integral FunctionsChelsea New York NY USA 1949

[8] W Bergweiler ldquoOn the Nevanlinna characteristic of a compos-ite functionrdquo Complex Variables vol 10 no 2-3 pp 225ndash2361988

[9] W Bergweiler ldquoOn the growth rate of composite meromorphicfunctionsrdquoComplex Variables vol 14 no 1ndash4 pp 187ndash196 1990

[10] I Lahiri and D K Sharma ldquoGrowth of composite entire andmeromorphic functionsrdquo Indian Journal of Pure and AppliedMathematics vol 26 no 5 pp 451ndash458 1995

[11] S K Datta T Biswas and C Biswas ldquoMeasure of growth ratiosof composite entire and meromorphic functions with a focuson relative orderrdquo International Journal ofMathematical Sciencesand Engineering Applications vol 8 no 4 pp 207ndash218 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of