DESCRIPTION OF ZERO SEQUENCES FOR HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF FINITE $\lambda$-TYPE IN A CLOSED HALF-STRIP Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том б. № 2 (2014). С. 123-127.

УДК 517.53

DESCRIPTION OF ZERO SEQUENCES FOR HOLOMORPHIC AND MEROMORPHIC FUNCTIONS OF FINITE A-TYPE IN A

CLOSED HALF-STRIP

Abstract. We describe the zero sets of holomorphic and meromorphic functions f of finite A-type in a closed half-strip satisfying f (a) = f (a + 2ni) on the boundary.

Keywords: holomorphic function, meromorphic function, function of finite A - type, sequence of finite A-density, A-admissible sequence

Mathematics Subject Classification: 30D35

1. Introduction

Let f be a meromorphic function in the closure of the half-strip

S = {s = a + it : a > 0, 0 < t < 2n}.

Suppose f has neither zeros nor poles on dS, and f (a) = f (a + 2ni), a > 0. Denote by {sj} the zero sequence of function f in S, Sj = aj + itj, by {pj} the sequence of its poles in S.

Let S * be the strip S with the straight slits {raj + itj}, {t Re pj + i Im pj}, 1 ^ t < to.

Given So E S*, suppose log f (s0) is well-defined and let

The following Lemma is a counterpart of Jensen-Littlewood Theorem ([1]).

Н.Б. Сокульскля, Описание множеств нулей голоморфных и мероморфных в замкнутой полуполосе функций конечного А-типл.

© Сокульскля Н.Б. 2014.

Поступила 24 января 2014г.

N.B. SOKULSKA

(1)

where integral is taken along a piecewise-smooth path in S* U dS, which connects s0 and s.

By n(n, f) we denote the counting function of poles of f in the rectangle Rn = {a + it : 0 < a ^ n, 0 ^ t < 2n}. We let

0

and

(3)

0

Lemma 1. [2] Let f be a meromorphic function in the closure of half-strip S,

f (a) = f (a + 2ni), a > 0. Then

N(^ 1) - N(a, f) = c0(— f) - —C0(a0, f) + (— - l)Co(0, f), f a0 a0

a > a0 > 0. (4)

The Nevanlinna characteristic of such functions was defined in [2] as

T (a, f) = m0(a, f) - — m0(a0, f) + ( — - A m0(0, f) + N (a, f), a > a0 > 0,

a0 a0

where

2n

m0(a, f) = 2n /log+|f (a +it)|dt

0

Definition 1. A positive non-decreasing continuous unbounded function A (a) defined for all a > a0 > 0 is said to be a growth function.

Definition 2. Let A (a) be a growth function and f be a meromorphic function in S, such that f (a + 2ni) = f (a), a > a0 > 0. We say that f is of finite A-type if T (a, f) ^ AA(a + B), a > a0 for some constants A > 0, B > 0 and all a, a > a0 > 0.

We denote by A the class of meromorphic functions of finite A-type in S and AH the class of holomorphic functions of finite A-type in S.

In this paper we describe the zero sequences of holomorphic functions in AH, as well as zero and pole sequences of meromorphic functions in A.

For entire and meromorphic in C functions similar problems were solved by L. Rubel and B. Taylor ([3]), for holomorphic and meromorphic functions in a punctured plane the same was done by A. Kondratyuk and I. Laine ([4]).

2. Description of zero sequences of holomorphic and meromorphic functions of finite A-type in a half-strip

Let Q = {sj} be a sequence of complex numbers in S. By n(r/,Q) we indicate the counting

function of Q in the rectangle Rn and we let

a

N (a, Q) = J n(n, Q)drj.

0

Definition 3. A sequence Q = {sj} from S has a finite A- density if

N (a, Q) ^ AA(a + B)

for some positive constants A, B and all a, a > a0 > 0.

Definition 4. A sequence Q = {sj} from S is said to be A-admissible if it has finite A-density

and there are positive constants A, B such that

l k

for all a\, a2, a0 ^ ai < a2 and each k E N.

E

a\<ReSj ^a2

3fca 1

3fca2

k

l

eSj

Denote

2n

l

Ck (а, f) = 2П I — ikt log If (а + it)|dt, k Є Z.

For a meromorphic in R0 function f such that f (а) = f (а + 2ni) the following relations hold true (see [2]):

—kа ——kа

Ck(а, f )= T^k (f ) - -2k-

l

+iz

a-k(f)

k

C-k(а, f) =Ck(а, f) k Є N,

k

eSj

l

2k

Е

Pj G Ra

—Pj

where Sj, pj are its zeroes and poles in Ra respectively, and

2n

a(f>=2^/e-ik‘mdt- k E N

0

Theorem 1. A sequence Q in S is the zero sequence of the function in AH if and only if it is A-admissible.

Proof. Let Q = {sj} be the zero sequence of a function f from AH. Then by (6)

Ck(а2, f) Ck(аі, f) ak—'^2 - а—— ^ + l

;зkа2

з'оі

2k—'^2

ak—kk°l - a_ k—■l

2k—'^2 0kаl

Е

а2

Е

—SAk —а2 /

a-k

IF

+

2k—kаl ll

2k—kаl

Е

Е

—SA k —'^ /

—2kаl —2kа2

l

+ 2k

l

l

l

2k—kаl

Е

—SA k_ l

—'0i / 2k—'kа2 V —а2 /

/ o.cP v /

(—Sj )k Z—✓ (—Sj )k

V 7 SjGRal V 7

~^\ k

z

j

where О < а1 < а2. Then we obtain l k

Е

аl<Re Sj ^а2

l 2Ck(а2, f) 2Ck(аі, f) + a-k

(—Sj )k

+

;3kа2

k—'02

Е

Sj GRa 2

—'0i ' k

k

—2'o2 —2'oi

+

а2 / k—k°‘l

Е

Sj GRal

—0і

We have

Е

< Е Е l < п(аі + l, f) < N (аі + l, —) < A^^j + l + Bi), а^ Є Ro4 , i — l, 2,

f

for some constants Al,Bl > О.

k

а

e

e

eSj

а

а

e

e

Sj GRa

k

—SJ

s j GRa2

sj GRa2

k

eSj

sj GRal

sj GRal

s j GRa2

Sj GRal

l

l

k

l

k

eSj

а

e

Sj GRa

Sj GRa

We also get the estimate for the left-hand side of identity (7):

E

ai<Re Sj ^a2

eksj

^ A2A(a2 + B2) + A2A(ai + B2) + |a-k1 ^ eka2 ekai k

+

+

keka2

E

ea2

+

kekai

E

eai

g2ka2 g2kai

k

^ A2A(a2 + B2) + A2A(ai + B2) + c

e

ka2

e

kai

+

e2ka2 e2kai

+ keka2 N(a + 1-)) + kekai N(ai + 1’1

^ AA(a2 + B) + AA(ai + B)

ka2

kai

k E N, a2 > ai ^ a0,

l

l

l

k

l

S

l

S

e

e

sj GR^2

l

where A = max{Ai, A2, C}, B = max{Bi + l, B2}.

Theorem 2 in [2] implies that the sequence Q has a finite A-density. Hence, it is A-admissible. Let now Q = {sj} be A-admissible. Then the sequence Z = {zj}, Zj = eSj E C, is Ai-admissible in C, where Ai(r) = A(log r). By the Rubel-Taylor Theorem [3, p. 84], (see also [5, p. 29]), there exists an entire function F(z) of finite Ai-type with zero sequence Z = {zj}. Therefore, the function f (s) = F(es) is holomorphic of finite A-type in S with the zero sequence

{sj}. □

Theorem 2. A sequence Q in S is the zero sequence of a function in A if and only if it has finite A-density.

Proof. If Q = {sj} is the zero sequence of a function f, f E A, then from [2], we have

N (a, Q) = N (a, f ^ T (a, f) ^ BA(a + C),

for all a ^ a0 > 0 and some B, C > 0.

Let now Q = {sj} be a sequence of finite A-density. Then the sequence Z = {zj}, Zj = esj, has the finite Ai-density if Ai(r) = A(logr). By the Rubel-Taylor Theorem [3, p. 88] (see also [5, p. 35]) there exist a meromorphic function F of finite Ai-type with zero sequence Z. The function f (s) = F(es) is the meromorphic of finite A-type in S with zero sequence {sj}. □

Corollary 1. A sequence P = {pj} is the pole sequence of a function f from A if and only if it has finite A-density.

Proof. Apply Theorem 2 to the function i.

□

BIBLIOGRAPHY

1. J.E. Littlewood On the zeros of the Riemann zeta-function, Proc. Camb. Philos. Soc. 22 (1924), 295-318.

2. N.B. Sokul’s’ka Meromorphic functions of finite X-type in half-strip // Carpathian Mathematical Publications 2012. V.4, №2. P. 328-339 (in Ukrainian).

3. L.A. Rubel, B.A. Taylor Fourier series method for meromorphic functions // Bull. Soc. Math. France 1968. 96. P. 53-96.

4. A. Kondratyuk, I. Laine Meromorphic functions in multiply connected domains // Fourier series method in complex analysis (Merkrijarvi, 2005),Univ. Joensuu Dept. Math. Rep. Ser., 1G (2006), P. 9-111.

5. A.A. Kondratyuk Fourier Series and Meromorphic Functions. L’viv.: Izdat. L’viv Univ., 1988. 196 p (in Russian).

Наталья Богдановна Сокульская,

Львовский национальный университет имени Ивана Франко, ул. Университетская, І,

Т9000, г. Львов, Украина

E-mail: natalya.sokulska@gmail. com