arxiv:math/0502433v1 [math.cv] 21 feb 2005 · din, a. a. kondratyuk and ya. v. vasyl’kiv for a...

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21

Feb

2005 The representation of a meromorphic function

as the quotient of entire functions and

Paley problem in Cn: survey of some results

B. N. Khabibullin ∗

Dedicated to Professor N. I. Akhiezer in the year of his 100th anniversary

Abstract

The classical representation problem for a meromorphic function fin C

n, n ≥ 1, consists in representing f as the quotient f = g/h of twoentire functions g and h, each with logarithm of modulus majorized bya function as close as possible to the Nevanlinna characteristic. Herewe introduce generalizations of the Nevanlinna characteristic and givea short survey of classical and recent results on the representation ofa meromorphic function in terms such characteristics. When f has afinite lower order, the Paley problem on best possible estimates of thegrowth of entire functions g and h in the representations f = g/h willbe considered. Also we point out to some unsolved problems in thisarea.

Introduction

Let f be a meromorphic function (we write f ∈ Mer) in the complexplane. At the beginning of the last century the definition of the classicNevanlinna characteristic T (r; f) and the Liouville Theorem on the growthof entire functions were giving an one of first results on the representation ofa meromorphic function as the quotient of entire functions.

Theorem 0.1 (J. Liouville–R. Nevanlinna). A meromorphic function fin the complex plane can be represented as the quotient f = g/h of polinomialsg and h, i.e.

log(

|g(z)| + |h(z)|)

≤ O(log r) , r → +∞ ,

∗Supported by the Russian Foundation of Basic Research under Grant No. 00-01-00770.

1

http://arxiv.org/abs/math/0502433v1

THE REPRESENTATION OF A MEROMORPHIC FUNCTION... 2

iffT (r; f) ≤ O(log r) , r → +∞ .

The classic Nevanlinna Theorem on the representation of a meromorphicfunction f of bounded characteristic T (r; f) ≤ const , r < 1, in unit diskD as the quotient of bounded holomorphic functions in D can be consideredalso as one of sources of our survey.

A first essential result for transcendental meromorphic functions followsimmediately from the Lindelof Theorem of the beginning of XX century (see,for example, [Ru2], [GO]):

Theorem 0.2 (E. Lindelof). A meromorphic function f in the complexplane can be represented as the quotient f = g/h of entire functions g and hof finite type for order ρ, i.e.

log(

|g(z)| + |h(z)|)

≤ O(rρ) , r → +∞ ,

iffT (r; f) ≤ O(rρ) , r → +∞ .

Here we consider far-reaching generalizations of these classic results andgive a survey of basic and recent results on the problem of the representationof a meromorphic function in Cn as the quotient of entire functions in termsof various characteristics. Also we point out to some unsolved problems inthis area.

Let f ∈ Mer in Cn and

f =gfhf

(0.1)

be a canonical representation of f as the quotient of entire functions gfand hf which are locally relatively prime, i.e. at each point z ∈ Cn, wheregf(z) = hf(z) = 0, the functions g and h are relatively prime in the ringof germs of functions analytic at z. In Cn such representations always exist[GR], [H1]. Both functions

uf = max{

log |gf |, log |hf |}

(0.2a)

or

uf = log(

|gf |2 + |hf |

2)1/2

(0.2b)

are plurisubharmonic on Cn. Consideration of these characteristic functionsuf is useful and natural, since uf is used to define the various types of Nevan-linna characteristic in particular in the Shimizu-Ahlfors form etc. [GO], [HK],[Ko], [Ku2], [Kh4], [Sk1]–[Sk2], [St1]–[Ta].


A function g on Cn is said to be circular if g(eiθz) = g(z) for all z ∈ Cn

and θ ∈ R.First we introduce the circular Nevanlinna characteristic

T cf (z) =

1

2π

2π∫

0

uf(eiθz) dθ , z ∈ Cn , (0.3)

which is a circular plurisubharmonic function [Ku2]. For example, if f(0) = 1and Tf(t; ζ), t ≥ 0, are the Nevanlinna characteristics (in the Shimizu-Ahlforsform for the case (0.2b)) [GO] of the family of the meromorphic functionsfζ(w) = f(wζ) of one variable w ∈ C, where ζ runs through the unit sphereSn ⊂ Cn, then Tf(t; ζ) ≤ T c

f (z), z = teiθζ . Equality holds if n = 1 or if gfand hf do not have common zeros (n > 1).

Let L = (L1, . . . , Lk) be a fixed simply ordered collection of complexvector subspaces in Cn and Cn is the direct sum of Lk. As usual, we write

Cn = L1 ⊕ · · · ⊕ Lk . (0.4)

Denote by R+ the set of nonnegative numbers.Let z ∈ Cn and

z = w1 + · · · + wk, wp ∈ Lp, p = 1, . . . , k, (0.5a)

rp = |wp|, ~r = (r1, . . . , rk) ∈ Rk+, (0.5b)

where |wp| is the Euclidean norm wp in Lp. For every z ∈ Cn the represen-

tation (0.5a) is unique.If u(z) is a function on Cn, then we consider the function u also as the

function u(z) = u(w1, . . . , wk) on the direct sum (0.4) under the notations(0.5). We keep also the same notation u for u(w1, . . . , wk).

We introduce Nevanlinna L-characteristic of f ∈ Mer in the form

T L(~r ; f) =

∫

S1

· · ·

∫

Sk

uf(r1ζ1, . . . , rkζk) ds(1)(ζ1) · · · ds(k)(ζk), ~r ∈ R

k+ ,

where the functions uf are defined in (0.2), and ds(p) is the area element onthe unit sphere Sp in Lp.

The Nevanlinna L-characteristic can be defined also in the form

T L(~r ; f) =

∫

S1

· · ·

∫

Sk

T cf (r1ζ1, . . . , rkζk) ds(1)(ζ1) · · ·ds

(k)(ζk). (0.6)


It is easy to check that every Nevanlinna L-characteristic is defined by(0.1)–(0.2) to within an additive constant.

The Nevanlinna L-characteristic gives the various variants of the classicalNevanlinna characteristics of f in Cn. So, if k = 1 and L1 = Cn in (0.4), thenwe get Nevanlinna characteristic T (r; f), r ≥ 0, by exhaustion of Cn by ballsrBn, where Bn is the unit ball in Cn; if k = n and L1 = · · · = Ln = C in (0.4),then we get Nevanlinna characteristic T (~r ; f), ~r ∈ Rn

+, by exhaustion of Cn

by polydisks P n(~r ) = {z ∈ Cn : |wp| = rp, p = 1, . . . , n} with the preceding

notations (0.5) (see [St1]–[Ta], [Ku2], [Sk1]–[Sk2], [Kh4] for various variantsof definitions of such Nevanlinna characteristics).

Below we formulate the well-known and recent main results on the rep-resentation of a meromorphic function on Cn, n ≥ 1.

The author expresses deep gratitude to I. F. Krasichkov-Ternovskiı foruseful discussion of this review and a correction of the text, and to M. L. So-din, A. A. Kondratyuk and Ya. V. Vasyl’kiv for a valuable information, andalso to the reviewer for helpful remarks.

1 The representation of arbitrary

meromorphic functions

A continuous increasing nonnegative function λ on R+ is called a growthfunction (see [RT], [Ml2] and [Ku2]).

For n = 1 and for a meromorphic function f of finite λ-type, i.e. satisfyingthe condition

T (r; f) ≤ const ·λ(r) + const , r > 0 , (1.1)

where λ has either the slow growth, i.e. satisfies the condition

λ(2r) ≤ O(

λ(r))

, r → +∞ , (1.2)

or log λ(exp t) is convex , L. A. Rubel and B. A. Taylor [RT, Theorems 5.4,3.5-6, 3.2] indicated (1968) a construction of entire functions g and h suchthat f = g/h and

log(

|g(z)| + |h(z)|)

≤ Aλ(Br) + C , r = |z| > 0 , (1.3)

where A, B and C are constants .All conditions for λ was removed for n = 1 (1970-72) by J. B. Miles in

[Ml1], [Ml2] (see also the original proof of J. B. Miles in [Ko, Ch. 4], [Ru2,Ch. 14]):


Theorem 1.1 (L. A. Rubel–B. A. Taylor–J. B. Miles). Every functionf ∈ Mer in the complex plane C can be represented as the quotient f = g/hof entire functions g and h such that

log(

|g(z)| + |h(z)|)

≤ AT (Br; f) + C , r = |z| > 0 , (1.4)

where A, B and C are constants†.

Analogous results for functions in the half-plane and in the unit disk canbe find in [Mlt] and in [Beck] recp..

The following special case of the representation theorems was establishedfor n = 1 in [Gl] (1972):

Theorem 1.2 (A. A. Gol’dberg). Every meromorphic function f in thecomplex plane C can be represented as the quotient f = g1/g2 of entire func-tions g1 and g2 without common zeros such that

log T (r; gk) = o(

T (r; f))

, r → ∞ , k = 1, 2 .

This theorem (for functions g1 and g2 without common zeros) is, in acertain sense, best possible (see also [Gl]). Further results in this direction(without common zeros) can to find in [Sk3], [Be2], [VSh], [VK].

Our last one-dimensional result [Kh13] (2001) is the following

Theorem 1.3 (B. N. Khabibullin). Let f be meromorphic function inthe complex plane. For each nonincreasing convex function ε(r) > 0 in R+

entire functions g and h in the representation f = g/h can be chosen so thatthe estimate

log(

|g(z)| + |h(z)|)

≤Aε

ε(r)T(

(1 + ε(r)) · r ; f)

+Bε , r = |z| > 0 , (1.5)

holds. Here Aε and Bε are constants depending on the function ε(r) (butindependent of r).

In particular for 1 + ε(r) ≡ B > 1 in (1.5) the last Theorem 1.3 impliesthe Rubel–Taylor–Miles Theorem 1.1, and for ε(r) = ǫ

/

(1 + r) in (1.5)with a constant ǫ > 0 we obtain the sharpening of the one-dimensionalvariant of Skoda Theorem 1.6 with (1 + r) in place of (1 + r)3 in (1.7).Besides the Theorem 1.3 can be essentially stronger than the Rubel–Taylor–Miles Theorem or the one-dimensional Skoda Theorem if the Nevanlinna

†The importance of such results for n > 1 was marked by W. Stoll in his book [St, p. 85]:“ Unfortunately the theorem of Miles [Ml2] has not been proved for several variables”.


characteristic T (r; f) has rapid growth and the decreasing function ε(r) ischosen in conformity with the growth of T (r; f).

For n > 1 the first result on the representation problem for a specialgrowth functions λ(r) = rρ (that solves the problem of L. A. Rubel [Eh,Problem 8] (1968)) is apparently due to W. Stoll [St1, Proposition 6.1] (1968).

Let λ(~r ) = λ(r1, . . . , rn), ~r ∈ Rn+, be a positive continuous function which

is nondecreasing in each variable, and T (~r , f) ≤ λ(~r ), ~r ∈ Rn+. B. A. Taylor

posed the problem‡ [Ta, p. 470] (1968):

(TP) when can a meromorphic function f be represented as the quotientf = g/h of entire functions g and h such that

log(

|g(z)| + |h(z)|)

≤ Aλ(B~r ) + C , rj = |zj| ≥ 1 , 1 ≤ j ≤ n , (1.6)

where A, B and C are constants ?

He proved the following Theorem [Ta, Theorem] (1968):

Theorem 1.4 (B. A. Taylor). If λ(~r ) is slowly increasing in each variable,in the sense that

λ(r1, . . . , 2rj, . . . , rn) ≤ Ajλ(r1, . . . , rn)

for some constant Aj > 0, 1 ≤ j ≤ n, and T (~r ; f) ≤ const ·λ(~r ) for allrj ≥ 1, 1 ≤ j ≤ n, then there are entire functions g and h such that f = g/hand (1.6) holds for all rj ≥ 1.

Further progress for meromorphic functions f in several variables and forthe characteristic T (r; f), i.e. k = 1, under special conditions on the behaviorof λ, was made in [Ku1], [Ku2, Propositions 7.3, 9.10] (1969-71):

Theorem 1.5 (R. O. Kujala). If there are constants a0 (resp., a vanishingfunction a0 = a0(r), r → +∞), a, b and R in R+ and p0 in N such that

r∫

s

λ(t)t−p−1 dt ≤ a0λ(br)r−p + aλ(bs)s−p

whenever r ≥ s > R and p0 ≤ p in N (in particular, if λ satisfies theslow growth condition (1.2)), then under the condition (1.1) the meromor-phic function f in Cn can be represented as the quotient f = g/h of entirefunctions g and h such that (1.3) (recp., with a vanishing function A = A(r),r → +∞) is valid.

‡In the original, “When is Λ the field of quotients of ΛE ?”


Except the Theorems 0.1, 0.2, 1.2, 1.3 and the mentioned result for λ(r) =rρ from W. Stoll [St1], all these results were obtained by the Fourier seriesmethod (see [Ko], [Ku2], [Ml2], [No], [RT]–[Ru2], [Ta]). This method wasused first by N. I. Akhiezer in [A] (1927) for the new proof the LindelofTheorem and later by L. A. Rubel in [Ru1] (1961) (see in this connectionalso [GO, P. 85–88]).

Using the ∂-problem method, H. Skoda [Sk1]–[Sk2] (1971-72) obtainedthe following results (see also [St, Theorem 9.12]):

Theorem 1.6 (H. Skoda). For every constant ǫ > 0 entire functions g andh in representation f = g/h ∈ Mer can be chosen to satisfy the estimates

log(

|g(z)| + |h(z)|)

≤ C(ǫ, s)(1 + r)4n−1T (r + ǫ; f) (1.7)

orlog

(

|g(z)| + |h(z)|)

≤ C(ǫ, s)(

log(1 + r2))2T(

(1 + ǫ)r; f)

(1.8)

for all r = |z| ≥ s > 0, where C(ǫ, s) is a constant depending on ǫ and s.

It should be noted [St, p. 295] that the pair (g, h) in the Theorem 1.6 canhave a common divisor, can depend on ε but can not depend on s, and canbe different in (1.8) from the pair chosen in (1.7). The case (1.7) is good forrapid growth, the case (1.8) is good for slow growth.

An analysis of these results indicates a definite rift between the cases ofslow and rapid growth of the majorants λ or of the characteristics T (·; f),both in methods (the Fourier series method in the Theorems 1.1, 1.4–1.5 andthe ∂-problem method in the Theorem 1.6) and the estimates and conditionson λ or T (·; f). Our balayage (or sweeping out) method (1990-93) enablesus to get rid of this rift and yields results in a complete form.

To be more specific, first this balayage method was applied for a newshort nonconstructive proof of Rubel-Taylor-Miles Theorem 1.1 in [Kh1, § 3](1991). The improvement of this method for n = 1 made it possible recentlyto prove the Theorem 1.3. In our articles [Kh2], [Kh4] we established forfunctions of several variables the following results that are, in a certain sense,extreme relative to the Nevanlinna characteristics T (r; f), r ≥ 0, and (0.3).

Theorem 1.7 (B. N. Khabibullin). Let f ∈ Mer, n ≥ 1. Then

(T) [Kh2, Theorem 1.3], [Kh4, Theorem 5] (1992-93)

For each constant ε > 0 the function f can be represented as the quo-tient

f =g

h, g and h are entire functions, (1.9)

such that

log(

|g(z)| + |h(z)|)

≤ AεT(

(1 + ε)r; f)

+ Cε , r = |z| > 0; (1.10)


(Tc) [Kh4, Theorem 4] (1993)

Under the conditions T cf (z) ≤ λ(z), z ∈ C

n, f(0) = 1, where thefunction λ is circular continuous positive increasing on all rays withorigin at 0 and satisfies the following two Hormander conditions [H2]:

(L) log(

1 + |z|)

≤ O(

λ(z))

as |z| → +∞ , and

(H) for any number ǫ > 0 there exist positive constans c1, c2, c3, c4 suchthat

|z−ζ | ≤ exp(

−c1λ(z)−c2)

=⇒ λ(ζ) ≤ c3λ(

(1+ǫ)z)

+c4, z ∈ Cn,

or

(H) for any ǫ > 0 there exist positive constants σ, c1, and c2 such that

|z − ζ | ≤ σ =⇒ λ(ζ) ≤ c1λ(

(1 + ǫ)z)

+ c2, z ∈ Cn, (1.11)

for any ε > 0 the function f can be represented as a quotient (1.9) suchthat

log(

|g(z)| + |h(z)|)

≤ Aελ(

(1 + ε)z)

+ Cε , z ∈ Cn , (1.12)

where under the condition (1.11) functions g and h can be chosen sothat g(0) = h(0) = 1.

Here Aε and Cε are constants depending on ε.

Besides the fact that the extra conditions on the growth function λ wasremoved in the Theorem 1.7 (Part (T)), it also refines the Theorems 1.1,1.5 and 1.6. For example, the constant B in (1.3) and (1.4) is replaced by aconstant 1+ε > 1 arbitrary close to 1, and in (1.7) and in (1.8) it is possible toremove the power factor before T (·; f) if in (1.7) r+ε is replaced by (1+ε)r,where ε > 0. Following Gol’dberg [Gl], we can show that we cannot setε = 0 in general in (1.10) and in (1.12). An analog of the Theorem 1.7 (Part(T)) for δ-subharmonic functions was established by O. V. Veselovskaya [Vs](1984) by the Fourier series method.

If we have some additional information on the functions gf and hf in theinitial representation (0.1), then these additional properties can be preservedsometimes in the final representation (1.9) (see [Kh9, Theorems 1–3] (1998)):

Theorem 1.8 (B. N. Khabibullin). If it is known in addition to conditionsof Theorem 1.7 that for f ∈ Mer with f(0) = 1 there exists the representation(0.1), where gf and hf are bounded in some open set B ⊂ Cn, then in (T)


(recp., in (Tc)) functions g and h can be chosen so that besides (1.10) (recp.,(1.12)) for any γ ≥ 1 the inequlity

log(

|g(z)| + |h(z)|)

≤ Cγ log(

2 + |z|)

, z ∈ Bγ ,

holds, where

Bγ = {z ∈ B : dist(z, ∂B) ≥(

1 + |z|)−γ

} ,

dist(z, ∂B) is the distance from z to the boundary ∂B of B, and Cγ is con-stant.

If n = 1, then functions g and h in (1.9) can be chosen also so that g andh are bounded on the set {z ∈ B : dist(z, ∂B) ≥ 1/γ}.

For the general case of the Nevanlinna L-characteristic we have at presentthe following result [Kh11] (2000):

Theorem 1.9 (B. N. Khabibullin). Let f ∈ Mer and Cn is representedas (0.4). Then, with the preceding notations (0.2)–(0.6), for any constantε > 0, there exists a representation (1.9) such that

log(

|g(z)| + |h(z)|)

≤ AεT+

L

(

(1 + ε)~r +Bε ·~1 ; f)

+ Cε log(

2 + |z|)

(1.13)

for all z ∈ Cn, where Aε, Bε, Cε are constants and ~1 = (1, . . . , 1) ∈ Rk+,

T+ = max{T, 0}.

It is easy to see, that the Theorem 1.9 implies the Theorem 1.7 (Part(T)), because always log

(

2 + |z|)

≤ O(

T (r, f))

, r = |z|, f 6≡ 0,∞. Also, if

log(

2 + |~r |)

≤ O(

λ(~r ))

, k = n, |~r | =√

r21 + · · · + r2k , (1.14)

then the Theorem 1.9 is the extension of Theorem 1.4. It solves the Taylorproblem (TP) with minimal estimate (1.14) from below for λ(~r ).

A general scheme of the solution of the representation problem and someother problems (balayage method) was presented in [Kh7]–[Kh8], [Kh12].

Unsolved problems.

Problem 1. Is it true, that the summand C log(

2 + |z|)

in (1.13) in theTheorem 1.9 can be removed ?

In any case it is true for k = 1, and in the Theorem 1.4 for k = n if λ(~r )is slowly increasing.


Problem 2. Extend the result of the Theorem 1.3 to Cn, n > 1, for classicNevanlinna characteristic T (r; f) and for Nevanlinna L-characteristic.

Problem 3. Denote by PSHp the cone of all plurisubharmonic function onLp, p = 1, . . . , k, where Lp was defined in (0.4). Denote by M+

p the cone ofall positive Borel measures (Radon measures) with compact support in Lp .

We introduce in M+p the partial order ≺ by (see [M, Ch. XI])

( ν≺µ ) ⇐⇒(

∫

v dν ≤

∫

v dµ for all v ∈ PSHp

)

.

If ν ≺ µ, then the measure µ ∈ M+p , is called a balayage ( also sweeping out

or Jensen measure [Ga]) of the measure ν (with respect to the cone PSHp ).Further, let µ = (µ1, . . . , µk) be a fixed simply ordered collection of mea-

sures, where every measure µp ∈ M+p is the balayage of the Dirac measure

δp ∈ M+p at the point 0 ∈ Lp, i.e.

∫

u dδp = u(0).Let uf be a plurisubharmonic function from (0.2). A function

T L,µ(~r ; f) =

∫

S1

· · ·

∫

Sk

uf(r1ζ1, . . . , rkζk) dµ1(ζ1) · · ·dµk(ζk) , ~r ∈ Rk+,

will be called the Nevanlinna-Jensen ( L,µ)-characteristic of f ∈ Mer. Howcan the Theorem 1.9 be extended for the Nevanlinna-Jensen ( L,µ)-charact-eristic ?

2 The representation of meromorphic

functions with restrictions on the type

and on the circled indicator

In this section we consider the problem of the representation of a mero-morphic function f in Cn as a quotient (1.9) with best possible estimates ofthe circled indicators and of the types of the entire functions g and f .

The representation (or factorization) type σ∗n(f, ρ) of a meromorphic func-

tion f on Cn of order ρ is defined to be the infimum of the numbers σ forwhich there are entire functions g and h of order ρ with the type less than σ(see [LG]) such that f = g/h.

Let

P1(ρ) =

πρ

sin πρ, if 0 ≤ ρ ≤ 1/2 ,

πρ , if ρ > 1/2 .(2.1)


For n = 1 the first sharp result in this direction was established apparentlyin our paper [Kh3, Theorem 2] (1992):

Theorem 2.1 (B. N. Khabibullin). Let f be a meromorphic function inthe complex plane C. If the type of the Nevanlinna characteristic T (r; f) isequal to σ with order ρ, i.e.

lim supr→+∞

r−ρT (r; f) = σ , (2.2)

then the sharp estimate σ ≤ σ∗1(f, ρ) ≤ σP1(ρ) is valid.

For n > 1 we set

Pn(ρ) = P1(ρ)n−1∏

k=1

(

1 +ρ

2k

)

. (2.3)

For n > 1 we have [Kh5, Theorem 3] (1993):

Theorem 2.2 (B. N. Khabibullin). If the type of the Nevanlinna charac-teristic T (r; f) is equal to σ with order ρ, i.e. (2.2) holds, then

σ ≤ σ∗n(f, ρ) ≤ σP1(ρ)

n−1∏

k=1

b( ρ

2k

)

≤ en−1Pn(ρ),

where

b(x) =

{

ex , if x ≥ 1 ,

ex , if x < 1 ,(2.4)

and for ρ ≤ 1 the sharp estimate

σ ≤ σ∗n(f, ρ) ≤ σPn(ρ) (2.5)

is valid.

Let u be a plurisubharmonic function on Cn of finite type with order ρ.Define the function

hcρ(z, u) = lim supξ→∞

|ξ|−ρu(ξz), ξ ∈ C , z ∈ Cn .

Its upper semicontinuous regularization

h∗cρ(z, u) = lim supζ→z

hcρ(ζ, u)


is called [LG, Definition 1.29] the circled indicator of growth of u (with orderρ). If we are dealing with circled indicator of growth h∗cρ(ζ, u), then we as-sume a priori that u is of finite type with order ρ. It is known that the circledindicator is complex-homogeneous of degree ρ, i.e., h∗cρ(ξz, u) = |ξ|ρh∗cρ(z, u),ξ ∈ C, is plurisubharmonic, and h∗cρ(z, u) ≥ 0 if u 6≡ −∞. We remind thatthe circular Nevanlinna characteristic T c

f (z) of f ∈ Mer in (0.3) is plurisub-harmonic. Therefore, circled indicator of growth for T c

f is defined. In terms ofthe circled indicator of growth h∗cρ(ζ, T

cf ) we have the following best possible

result [Kh5, Theorem 5.1, Example 5.1] (1993):

Theorem 2.3 (B. N. Khabibullin). Suppose that f ∈ Mer, f(0) = 1, andk is a continuous circular function on Sn ⊂ Cn.

1. If h∗cρ(ζ, Tcf ) < k(ζ), ζ ∈ Sn , then there exist entire functions g

and h such that f = g/h, g(0) = h(0) = 1, and

max{

h∗cρ(ζ, g), h∗cρ(ζ, h)}

< P1(ρ)k(ζ), ζ ∈ Sn .

The constant P1(ρ) cannot in general be diminished, not even for anyone of the functions g and h.

2. If f = g/h is a representation of f as the quotient of entire functionsg and h, then

max{

h∗cρ(ζ, g), h∗cρ(ζ, h)}

≥ h∗cρ(ζ, Tcf ) , ζ ∈ Sn ,

and this estimate is sharp.

The main difficulty consists in the obtaining the upper estimates in theTheorems 2.1–2.3. The success here was achieved with the use of the balayagemethod (see the preceding section).

Unsolved problems.

Problem 1. Prove the upper estimate in (2.5) for all ρ. See the commen-tary to the last Problem in the next section.

Problem 2. Extend the results of the Theorems 2.2–2.3 to the Nevanlinna L-characteristic and to the the Nevanlinna-Jensen ( L,µ)-characteristic.


3 Paley problem

The subject of this section is close to the themes of the previous sectionsin both estimates and methods of proofs.

For a function u in Cn (recp., in Rm) with the range of values [−∞,+∞]or R ∪ {∞}, we set

M(r; u) = max{

u(z) : |z| = r} , u+(z) = max{

u(z), 0}

.

In 1932 R. Paley supposed that for any entire function f of order ρ ≥ 0the inequality

lim infr→+∞

M(

r; log |f |)

T (r; f)≤ P1(ρ) (3.1)

holds (see the definition of P1(ρ) in (2.1)).In (3.1) the equality is achieved, in particular, for the Mittag-Leffler’s

function (see, for example, [GO, p.111]). Earlier the inequality (3.1) wasproved by G. Valiron [V] (1930) and by A. Whalund [Wh] (1929) for ρ ≤ 1/2.

The Paley conjecture was conclusively proved by N. V. Govorov [Gv]only in 1967. In 1968, V. P. Petrenko [Pe] extended this result to meromor-phic functions of finite lower order. Different proofs was obtained later byI. V. Ostrovskiı [Os] for meromorphic functions and by M. R. Essen [Es] forsubharmonic functions in the complex plane.

Theorem 3.1 (N. V. Govorov–V. P. Petrenko). If f is a meromorphicfunction in the complex plane C of finite lower order λ, i.e.,

λ = lim infr→+∞

log T (r; f)

log |r|< +∞ , (3.2)

then the inequality

lim infr→+∞

M(

r; log |f |)

T (r; f)≤ P1(λ) (3.3)

holds.

The relative growth of T (r; f) and M(

r; log |f |)

for entire and meromor-phic functions of infinite order in the complex plane has also been consideredby many authors.

For entire functions of infinite order in the complex plane in [Ch](1981),[MSh](1984) and finally in [DDL](1990–1991) was proved the following result.


Theorem 3.2. (C. T. Chuang, I. I. Marchenko–A. I. Shcherba,C. J. Dai–D. Drasin–B. Q. Li) Let f be an entire function of infiniteorder in the complex plane. If ψ(x) is increasing and positive for x ≥ x0 > 0and if

∫ ∞

x0

dx

ψ(x)<∞ , (3.4)

then

lim infr→+∞

M(

r; log |f |)

T (r; f) · ψ(

log T (r; f)) = 0 (3.5)

and evenM

(

r; log |f |)

= o(

T (r; f) · ψ(

log T (r; f))

)

as r → +∞ through a set of logarithmic density one.

In [MSh] and [DDL] it is also shown that the results are best possible insome sense.

For meromorphic functions in [DDL] (1990–1991) the following was ob-tained.

Theorem 3.3 (C. J. Dai-D. Drasin-B. Q. Li). Let f be a meromorphicfunction of infinite order in the complex plane. If ψ(x) is the same as in theTheorem 3.2 then

M(

r; log |f |)

= o(

T (r; f) · ψ(

log T (r; f))

· logψ(

log T (r; f))

)

as r → +∞ through a set of logarithmic density one.

A different approach has been taken in [Be1], [BB] (1990–1994) where thecharacteristic M

(

r; log |f |)

has been compared with the derivative of T (r; f).

Theorem 3.4 (W. Bergweiler-H. Block). Let f be a meromorphic fun-ction of infinite order in the complex plane. Then

lim infr→+∞

M(

r; log |f |)

rT ′−(r; f)

≤ π ,

where T ′−(r; ·) is the left-side derivative of T of r.

Let ψ(x) be positive and continuously diffirentiable for x ≥ x0 > 0 suchthat ψ(x)/x is non-decreasing, ψ(x) ≤

√

ψ(x), and (3.4) is satisfied. Then(3.5) holds.


The articles of I. I. Marchenko [Ma1]-[Ma2] contains much informationin particular on the growth of entire and meromorphic functions of infiniteorder.

We don’t know any results for entire and meromorphic functions of infiniteorder in Cn, n > 1, wich are analogs of the Theorems 3.2–3.4.

Let u be a subharmonic function in Rm, m ≥ 2, (resp., in Cn) and let|Sm−1| be the area of the unit sphere Sm−1 in Rm, ds is the area element onthe unit sphere S

m−1.

mq(r; u) =

(

1

|Sm−1|

∫

|Sm−1|

∣

∣u(rx)∣

∣

qds(x)

)1/q

, r > 0, 1 ≤ q < +∞.

Then Nevanlinna characteristic T (r; u) and M(r; u) are respectively

T (r; u) = m1(r; u+), M(r; u+) = m∞(r; u+) = lim

q→+∞mq(r; u

+).

In particular if f is a entire function in Cn then T (r; f) = T(

r; log |f |)

,m = 2n.

An analogue of (3.3) for subharmonic functions of finite lower order λ inRm, m ≥ 3, was obtained by B. Dahlberg [D, Theorem 1.2].

To be more specific suppose λ ∈ (0,+∞) is given. The Gegenbauerfunctions Cγ

λ are given as the solutions of the differential equation

(1 − x2)d2u

dx2− (2γ + 1)x

du

dx+ λ(λ+ 2γ) u = 0, −1 < x < 1 ,

with the normalization

limx→1−0

Cγλ(x) = Cγ

λ(1) =Γ(λ+ 2γ)

Γ(2γ)Γ(λ+ 1).

Put aλ = sup{

t : Cm−2

2

λ (t) = 0}

and define the function uλ in Rm, m ≥ 3,by

uλ(x) =

{

0 if x1 ≤ aλr

rλCm−2

2

λ (x1/r) if x1 > aλr ,

where x = (x1, . . . , xm) and r = |x|.The function uλ is subharmonic in R

m and the lower order of uλ is λ. Set

c(λ,m) = lim infr→+∞

M(r; uλ)

T (r; uλ).


Theorem 3.5 (B. Dahlberg). Let u be a subharmonic function in Rm,m ≥ 3, of finite lower order λ > 0. Then we have that

lim infr→+∞

M(r; u)

T (r; u)≤ c(λ,m) ,

and this estimate is best possible.

The generalization of the Theorems 3.1, 3.5 is also known.

Theorem 3.6 (M. L. Sodin [So] (1983)). Let u be a subharmonic functionof lower order λ in the complex plane. Then

lim infr→+∞

mq(r; u+)

T (r; u)≤ mq(Sλ), 1 < q ≤ +∞, (3.6)

where mq(Sλ) is the Lebesque mean of order q, 1 < q < +∞, of the function

Sλ(ϕ) =

πλ cosλϕ if |ϕ| ≤π

2λ,

0 ifπ

2λ< |ϕ| ≤ π,

for λ ≥ 1/2 and

Sλ(ϕ) =πλ cosλϕ

sin πλ, |ϕ| ≤ π,

for λ < 1/2, m∞(Sλ) = max{

Sλ(ϕ) : |ϕ| ≤ π}

. The inequality (3.6) issharp.

This result was extended to m ≥ 3 in [KTV, Theorem 1] (1995).Set

Qm−2

2

λ (θ) =

{

A(λ,m)Cm−2

2

λ (cos θ) if 0 ≤ θ ≤ αλ,

0 if αλ < θ ≤ π,

where

αλ = min{

θ ∈ (0, π) : Cm−2

2

λ (cos θ) = 0}

,

A(λ,m) =

(

|Sm−2|

|Sm−1|

αλ∫

0

Cm−2

2

λ (cos θ)sinm−2θ dθ

)−1

.

Theorem 3.7 (A. A. Kondratyuk–S. I. Tarasyuk–Ya. V. Vasyl’kiv).Let u be a subharmonic function of lower order λ > 0 in Rm, m ≥ 3. Thenfor every q, 1 < q ≤ +∞, the inequality

lim infr→+∞

mq(r; u+)

T (r; u)≤Mq(λ,m) (3.7)


is true, where Mq(λ,m) = mq

(

Qm−2

2

λ

)

. There exists a subharmonic functionu of lower order λ > 0 in Rm for which in (3.7) the equality is achieved.

Besides, from W. Hayman result [HK] it follows that for subharmonicfunctions of order λ = 0 the inequlity

lim infr→+∞

mq(r; u+)

T (r; u)≤ 1, 1 < q ≤ +∞ ,

holds.In [Kh6] (1995) we extended (3.3) to meromorphic functions f in Cn,

n > 1. Instead of M(r; log |f |) the corresponding formulation involves in thiscase the family of characteristics M(r; log |fζ|), ζ ∈ Sn, of the slice functionsfζ(w) = f(wζ), w ∈ C. This is understanble, for in the case of a meromorphicfunction the quantity M(r; log |f |) can be identically equal to ∞ starting fromsome value of r, whereas the Nevanlinna characteristics of f can have veryslow growth (one example is the function f(z1, z2) = (1 + z1)/(1 + z2) in C2).

Theorem 3.8 (B. N. Khabibullin). Let f be a meromorphic function inCn of finite lower order λ, i.e., (3.2) holds. Then

lim infr→+∞

M(

r; log |fζ|)

T (r; f)≤ Pn(λ) , ζ ∈ Sn, (3.8)

where Pn(λ) was defined in (2.3), and this estimate is best possible.

The estimate (3.8) was new also for entire functions and was not impliedin general by the above result of B. Dahlberg for subharmonic functions inRm, m ≥ 3.

On the other hand, one can pose the problem of finding a complete andbest possible analogue of (3.3) for plurisubharmonic functions and entirefunctions of several variables. Such result was obtained in our paper [Kh10,Theorem 1] (1999):

Theorem 3.9 (B. N. Khabibullin). Let u be a plurisubharmonic functionin Cn of finite lower order λ. Then

lim infr→+∞

M(r; u)

T (r; u)≤ Pn(λ) (3.9)

for λ ≤ 1 and this estimate is best possible. For λ > 1,

lim infr→+∞

M(r; u)

T (r; u)≤ P1(ρ)

n−1∏

k=1

b( ρ

2k

)

≤ en−1Pn(ρ), (3.10)

where b(x) was defined in (2.4).


More can be said about the definitive character of (3.9). For each ρ ≥ 0there exist in Cn entire functions of order ρ and normal type such that

limr→+∞

M(

r; log |f |)

T(

r; log |f |) = Pn(ρ) .

Comparing with the Dahlberg’s Theorem 3.5, we can draw the followingconclusions:

(a) for λ = 0 or λ = 1 the result of the Theorem 3.9 is a consequence ofDahlberg’s Theorem;

(b) for 0 < λ < 1 and n = 2 Theorem 3.9 does not follow from Dahlberg’sTheorem;

(c) for λ ≥ πe− 1 and n = 2 even the estimate (3.10) in Theorem 3.9 doesnot follow from Dahlberg’s Theorem;

(d) as λ → +∞, for n = 2 the estimate (3.10) is better by order O(1/λ)than the one following from Dahlberg’s Theorem.

Unsolved problems.

Problem 1. Extend the results of the Theorems 3.2–3.4 for entire andmeromorphic functions of several variables of infinite order.

Problem 2. Extend the results of the Theorems 3.8–3.9 to the Nevanlinna L-characteristic and to the the Nevanlinna-Jensen ( L,µ)-characteristic.

Problem 3. Extend the results of the Theorem 3.9 for the mq(r; ·) insteadof M(r; ·).

Problem 4. Prove the upper estimate in (3.9) for all λ.

Commentary. In order to prove the upper estimates in (2.5) for all ρand in (3.9) for all λ, it is sufficient to confirm the following:

Hypothesis. Let S be a nonnegative increasing function on R+, S(0) = 0,and the function S(t) is convex with respect to log t, i.e., S(ex) is convex on[−∞,+∞). Further, let λ ≥ 1/2, n ∈ N, n ≥ 2. If

1∫

0

S(tx)(1 − x2)n−2x dx ≤ tλ , 0 ≤ t < +∞ , (3.11)


then+∞∫

0

S(t)t2λ−1

(1 + t2λ)2dt ≤

π(n− 1)

2λ

n−1∏

k=1

(

1 +λ

2k

)

. (3.12)

This Hypotesis is true if λ ≤ 1. When

S(t) = 2(n− 1)n−1∏

k=1

(

1 +λ

2k

)

tλ , λ ≥1

2,

we have the equalities in (3.11) and in (3.12).It is not known even, whether the formulated Hypothesis is true for n = 2

and λ > 1 ?

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Department of Math., Bashkir State University, Frunze St., 32, Ufa, Bashko-rtostan, 450074, Russia

Institute of Math. of Ufa Scientific Center, Chernishevskiı St., 112, Ufa,Bashkortostan, 450025, Russia

E-mail: algeom@ bsu.bashedu.ru and(or) [email protected]

arxiv:math/0502433v1 [math.cv] 21 feb 2005 · din, a. a. kondratyuk and ya. v. vasyl’kiv for a...

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