Modulo-loxodromic meromorphic functions in ℂ \{0} Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 8. № 4 (2016). С. 156-162.

УДК 517.53

MODULO-LOXODROMIC MEROMORPHIC FUNCTIONS

IN C \

A.YA. KHRYSTIYANYN,

{0}

A.A. KONDRATYUK

Abstract. We introduce modulo-loxodromic functions and study their representations, zeroes and poles distribution. We also show that each modulo-loxodromic meromorphic function in C \ {0} is Julia exceptional.

Keywords: loxodromic meromorphic function, modulo-loxodromic function, Julia exceptional function.

Mathematics Subject Classification: 30D35, 30D45

1. Introduction

In the work [1, p. 133], which A. Ostrowski [2] called "besonders schone und überraschende", G. Julia gave an example of a meromorphic in the punctured plane C* = C \ {0} function such that

/ (qz) = f (z) (1)

for some non-zero q, |g| = 1, and all z G C*. He noted that the family {fn(z)}, fn(z) = f (qnz) is normal [3] in C* since fn(z) = f (z) for all z G C*.

The functions satisfying (1) are called multiplicatively periodic. The theory of meromorphic in C \ {0} multiplicatively periodic functions was developed by O. Rausenberger [4]. G. Valiron [5] (see also [6]) called these functions loxodromic since for non-real q, the points, at which such function takes the same value, are located on a logarithmic spirals (the images of loxodromes under the stereographic projection). They give a simple construction [5], [6] of elliptic functions, which are well known due to the works of K. Jacobi, N. Abel, and K. Weierstrass.

We consider modulo-loxodromic functions.

2. Modulo-loxodromic meromorphic functions

Definition 1. A meromorphic in C* function f is said to be modulo-loxodromic with a multiplicator q if there exists q (0 < |g| < 1) such that If (qz)I = If (z)|, z G C*.

We denote by |£|g and Cq the sets of all modulo-loxodromic and loxodromic functions with a multiplicator q, respectively.

It is obvious that Cq C |£|g. However, there are modulo-loxodromic functions in C* which are not loxodromic.

Indeed, consider an entire function with the zero sequence {q-n}, n G N, where 0 < |g| < 1,

<x

h(z) = n(1 - <lnz).

n=l

A.Ya. Khrystiyanyn, IN C \{0}.

A.A. Kondratyuk

Modulo-loxodromic meromorphic functions

© Khrystiyanyn A. Ya., Kondratyuk A. A. 2016. Поступила 21 октября 2015 г.

The function

P(z) = (1 - z)h(z)h(-\ = (1 - z) n(1 - qnz)(l -

\ / n=1 ^ '

is called the Schottky-Klein prime function [8].

This function is holomorphic in C* with zero sequence {qn}, n E Z. It was introduced by Schottky [9] and Klein [10] for studying the conformal mappings of doubly-connected domains.

Now consider the function

™ = PP^ ■ an* Q (2)

Taking into consideration that

P (qz) = - \P (z),

we have f(qz) = eiaf (z) and | f(qz)l = | f(z)\, z E C*. Hence, f E \C\q and f E £q.

Furthermore, f E £q for each q. Indeed, suppose that there exists a non-zero a, |a| = 1 such that f(az) = f(z) foreach z E C*. We observe that f(eta) = 0. So, f(aeia) = 0 and since the only zeroes of f are qke%a, k E Z, we obtain that there exists k0 E Z such that a = qk°. This implies that f(az) = f(qk° z) = elk°af (¿) and the last value cannot be equal to f(z) for any k0 E Z due to the choice of a in (2).

3. Representation of modulo-loxodromic functions

Let f be a meromorphic in C* modulo-loxodromic function with a multiplicator q.

First, we suppose that f is holomorphic in C*. Then f is bounded in a neighbourhood of the origin since Ifl is determined by its values in Aq = {z E C : |g| < Izl ^ 1}. Hence, the origin is a removable singularity of . We have that is holomorphic and bounded in C. Therefore, by the Liouville theorem = n .

If is not holomorphic, then there exists at least one pole E C* of . Taking into consideration the modulo-loxodromity of f, we conclude that qnb, where n E Z are also the poles of f. Applying similar arguments to 1/f, we obtain that f is either constant or has at least one zero a E C*. In the latter case (fa, where n E Z must also be the zeroes of f.

Thus, we have only two mutually excluding possibilities. Either function f is constant or it has an infinite number of zeroes and poles.

The function log Ifl is loxodromic i-subharmonic function [7]. Applying [7, Thm. 3.3], we conclude that the function has the same number of zeroes and poles (taken counting multiplicities) in each annulus {z : Iqlr < Izl ^ r}, r > 0.

Let a\, a2,...,am and bb2,..., bm be the zeroes and poles of f in {z : |g| < Izl ^ 1}, respectively. Then all the zeroes of f have the form akqn, where n E Z, k = 1, 2,... ,m, while all the poles of f are given by bkqn, where n E Z, k = 1, 2,... ,m, and there exists p E Z such that

^tH = lqlp. (3)

\bi--- bml

A Nevanlinna type characteristic of a function meromorphic in {z : ^ < \z\ < Ro}, where 1 < R0 ^ was introduced in [11]. Namely,

To( r, f) = mo( r, f) + No( r, f), 1 <r<Ro,

where

mo( r, f) = m( r, f) + ^f ^ - 2m(1, f),

1 C ^ 1

m(t, f) = ~ log+ I f(tei&)\dd, — < t < Ro, 2n Jo Ro

and

Na(r, f)

n0 (t, /)

dt,

Ji t

n0(t, f) is the number of poles of f in the annulus 1/t < \z\ ^ t taken counting multiplicities.

The function T0 (r, f) is nonnegative, nondecreasing, continuous and convex with respect to logr on [1, R0) ([11]). By the definition we have T0(r, 0) = 0.

Taking into consideration the above observations on zeroes and poles of f and analysing the proof of Theorem 1 in [12], we obtain that the statement of this Theorem 1 is valid for our function . That is,

To(r, f)

m

log ill

Y log r + O(logr),

r > 1.

So, the function f is of order 0 and applying the representation theorem for a meromorphic in C* function of finite order ([13]), we get the following representation of f.

Theorem 1. Each modulo-loxodromic function f has the representation

,f( z) = Cz" n

„ n (1 - (1 -

n=0 \ ) ra=l V Z

n=0

k=^ (1 -Ç) n

n=l

1

qnbk

zeC*

where C is a constant and p G Z satisfies condition (3).

4. Zero and pole distribution Let [a,j}, {bj}, j G Z be a pair of sequences in C*, p G Z. Denote

M"(r)

n i«^. |

l<|aj

n

r > 1;

l<| bj l^r

n la^

r<l«j |<l

0 <r < 1.

n ^

r<\bj

Theorem 2. The sequence of zeroes {aj} and the sequence of poles {bj} of a modulo-loxodromic function satisfy the following conditions (i) the number of aj and bj in each annulus of the form {z : r < \z\ < 2r}, r > 0 is bounded

by an absolute constant; (ii) the difference between the numbers of aj and bk in each annulus {z : r1 < \z\ < r2}, 0 < r1 < r2 < is bounded by an absolute constant;

(iii) there exists C1 > 0 such that ^ — 1 > C1 for all j, k G Z;

(iv) the function Mp(r), where p G Z satisfies condition (3), is continuous and bounded for

> 0.

Proof. Let f be a modulo-loxodromic function. As we have established, either function f is constant or it has infinitely many zeroes and poles. The proof of the theorem in the former case is trivial, so we can focus only on the latter one. Next we use the representation (4).

(i) First we remark that there exists a unique n0 G Z+ such that , ,, ^ 2 < t^—^t. This

|g|n°+l

n0 is equal to

log 2

log ill

r

"

"

Since each annulus zeroes of f, say m, and

w <|z| ^

}

where k E Z, contains the same number of

(r, 2 r]

(S (i #, №k+i 0 u(\ ^

2

it follows that the annulus {z : r < I zl ^ 2r} contains at least n0m and less than (n0 + l)m zeroes of f. The same is true about the poles of f.

(ii) In the same way as in (i), we can find unique n\,n2 E Z such that

lqlni+l < ri ^ lqln1 < Iqp < r2 ^ \qp-l.

Hence,

(r i, r2) = (ri, \q\ni] U

/n2-l \

U (lvlk, lvlk+l]) u

\k=ni /

2).

Each annulus of the form {z : \q\k+l < \z\ ^ lqlk}, where k E Z, contains the same amount of zeroes and poles of f taken counting multiplicities; we have denoted this number by m. Therefore, the difference between the numbers of zeroes and poles of f in the annulus {z : ri < l z\ < r2} is no greater than 2m because of the choice of n, n2.

(iii) Let a\,a2,..., am and bb2,..., bm be the zeroes and the poles of f in {z : \g\ < \z\ ^ l}, respectively. Then all the zeroes of f have the form a^>k = akq^, where ^ E Z, k = l, 2,... ,m. The same is true about the poles of f, namely = bkqv, where v E Z, k = l, 2,... ,m. So, ^ = P-ql, where I E Z. '

We need to show that there exists C > 0 such that the inequality

Jji 1 1

-1

k

> C

holds for all , E {l, 2, . . . , m}, and E Z.

Suppose that for each > 0 there exist , k E {l, 2,

, m}, and E Z such that

•^i l 1 -TV - l

k

(5)

Without loss of generality we can assume that \/\ ^ 2. Indeed, taking into consideration the location of aj, bk, we have

ij i q

In the same way we get

k

aj i k

< l\q\l < \q\, 2.

> \q\lq\1 , K -2.

Thus, for all j,k E {l, 2,..., m}, and I ^ 2

1 1 -T? -1

k

and for I < — 2

Let now 1/1 < 2. Choose

l

Jji 1 1 -1

k

l l

^ l -\q\,

l

s R-L

e = - min{\ajql - bk \ : j,k E {l, 2,... ,m},-l ^ l}.

Then (5) implies That is,

|ajq1 - bk| ^ elbk| ^ e.

K^ - bk| ^ 2minilai^ - h| : j,k G {1, 2,..

.,m},-1 ^ I ^ 1}

which gives a contradiction.

(iv) We recall that f satisfies representation (4). Clearly, we can assume that C = 0. Consider

2t

the integral means I(r) = -If log | f(r elS )| dd , r > 0

Let z = re% . We have for r > 1 [16, p. 8]

2t

2* l0g

1

j

d9 = log+ — laj 1

and, if |aj| ^ 1,

2 tt

l0g

i -

de = 0.

The same is true for bj.

Thus,

i{r)=piogr + y^ log+ ^ - log+ + loglc 1, r>1.

laj |>1

lbj |>1

l ^ \

In the same way for 0 < r ^ 1 we obtain

I(r) = plogr + ^ log+ ^ - ^ log+ ^ + log \ C\ |aj |<1 r lbj |<1 r

Hence,

MP(r) = —— exp/(r), r> 0 . \^ \

(6)

Since I(r) is convex with respect to logr and therefore, is continuous, I(r) is bounded on [|g|, 1]. It follows from the definition of a modulo-loxodromic function that /(|g|kr) = I(r) for each k G Z. Then, we conclude that I(r) remains bounded for all r > 0 that completes the proof.

□

5. JULIA EXCEPTIONALITY

Definition 2. A meromorphic in C* function f is called Julia exceptional [3] if for some q, 0 < |g| < 1, the family {fn(z)}, n G Z, where fn(z) = f(cf1 z), is normal [3] in C*.

The following theorem is a generalization of one "remarkably complete" result of A. Ostrowski [2], [3] for meromorphic functions f with two essential singularities. This theorem was originally formulated without proof by A. Eremenko [14] and later was proved by L. Radchenko [15]. We propose here its following version.

n I h- la i1 - 7) n ( K' l>i 'l

n 1 lbj |<1 (' - * : I n 1 № l>i V1 -

Theorem A. I. Two sequences [a,j}, {bj} in C* are sequences of zeroes and poles of a Julia exceptional in C* function f, respectively, if and only if they satisfy conditions (i) -(iii) of Theorem 2 and

(iv) there exist p E Z and C2 > 0, C3 > 0 such that Mp(\aj|) ^ C2 and Mp(\bj|) ^ C3 for each j E Z .

II. If {aj}, {bj}, and p satisfy (i) - (iv), then the function

U{z) = zp

is Julia exceptional in C*, and vice versa, each non-rational Julia exceptional in C* function f satisfies the representation

f(z) = C • n(z)

where {aj}, {bj}, p satisfy (i) - (iv), and C is a constant.

As an immediate consequence of Theorem 2 and Theorem A [2], [3] we obtain the following theorem.

Theorem 3. Each modulo-loxodromic function is Julia exceptional in C*.

Indeed, using the representation of modulo-loxodromic function given by Theorem 1, we observe that conditions (i)-(iii) in Theorem A coincide with those of Theorem 2 and condition ( ) of Theorem A is implied immediately by condition ( ) of Theorem 2.

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Andriy Yaroslavovych Khrystiyanyn, Ivan Franko National University of Lviv, 1 Universytetska Str., 79000 Lviv, Ukraine E-mail: [email protected]

Andriy Andriyovych Kondratyuk , Ivan Franko National University of Lviv, 1 Universytetska Str., 79000 Lviv, Ukraine E-mail: [email protected]