UNIQUENESS THEOREMS FOR MEROMORPHIC FUNCTIONS ON ANNULI Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 12. № 1 (2020). С. 115-121.

UNIQUENESS THEOREMS FOR MEROMORPHIC FUNCTIONS ON ANNULI

A. RATHOD

Abstract. In this paper, we discuss the uniqueness problems of meromorphic functions on annuli. We prove a general theorem on the uniqueness of meromorphic functions on annuli. An analogue of a famous Nevanlinna's five-value theorem is proposed. The main result in this paper is an analog of a result on the plane C obtained bv H.S. Gopalkrishna and Subhas S. Bhoosnurmath for an annuli. That is, let f1(z) and f2(z) be two transcendental

meromorphic functions on the annulus A = jz : < |z| < E0 j, where 1 < R0 ^ Let

a,j, j = 1, 2,... ,q), be q distinct complex numbers in C, and kj, j = 1, 2,... ,q be positive integers or to satisfying

k\ ^ k2 ^ ... ^ kg.

If

Ekj)(%■, /1) = Ek.){aj, f2), j = 1 2,..., q,

and

Q

y^l---^ > 2,

^kj + 1 ki + 1 3=2

then fi(z) = f2(z).

Keywords : Nevanlinna theory, meromorphic functions, annuli. Subject Classification: 30D35

1. Introduction

The uniqueness theory of meromorphic functions is an interesting problem in the value distribution theory and as well as the uniqueness theory of algebroid functions. The uniqueness problem of algebroid functions was first considered by Valiron [3], afterwards several uniqueness theorems of algebroid functions in the complex plane C were proved. In 2005, A. Ya. Khrystivanvn and A. A. Kondratyuk have proposed on the Nevanlinna Theory for meromorphic functions on annuli (see [6], [7]) and after this work, many others work in this area appeared, see [5], [12], [13], [14], [15], [21], [28]). In 2009, Cao and Yi [8] studied the uniqueness of meromorphic functions sharing some values on annuli. In 2015, Yang Tan [2], Yang Tan and Yue Wang [1] proved some interesting results on the multiple values and uniqueness of algebroid functions on annuli and also others have proved several results for algebroid functions on annuli, see [9], [11], [16], [18], [19], [20], [22], [23], [24], [25], [26], [27], [29], [30]. Thus, it is interesting to consider the uniqueness problem of algebroid functions in multiply connected domains. By Doubly connected mapping theorem [10], each doubly connected domain is eonformally equivalent to the annulus {z : r < |z| < R], 0 ^ r < R ^ We consider only two cases : r = 0 R =

A. Rathod, Uniqueness theorems for meromorphic functions on annuli. © A. Rathod 2020 . Поступила 4 июня 2019 г.

simultaneously and 0 ^ r < R ^ +to. In the latter case the homothety z ^ ^ reduces the given domain to the annulus

R) = A I ,

Rn / R

A = A( Ro) = A ^ R, R^ = ^z: R < |z| < Rcj

where Rn = . Thus, in both eases, each annulus is invariant with respect to the inversion i

z M

z

2. Basic notations in the Nevanlinna theory on annuli

Let f be a meromorphie function on the annulus A = ^z : < |z| <Rn|, We recall classical notations of Nevanlinna theory as follows

N R f) = i" n(t ■ r> -"(0- f) it + n(0, f)iogR,

c

1 r2n

m(R,f) = 2^\ Xog+1f(ReW)idd,

T(R, f) = N(R,f)+m(R, f),

where log+ x = max[logx, 0}^d n(t, f) is the counting function of poles of the function f in [z : |z| ^ t}. Let

NiR f) = i1 ^dt, h 1

N2- f) = J* ^dt,

m (R, f) = m(R, f)+m(J-, f ) - 2m(l, f),

N(R, f) = Ni(R, f) + N2(R, f),

where n1(t, /^d n2(t, f) are the counting functions of the poles of the function f in [z : t < |z| ^ l^d [z : l < |z| ^ t}, respectively. The Nevanlinna charecteristic of f on the annulus A

To (R, f) = m0 ( r, f) + N0 (R, f).

Definition 1 (8). Let f(z) be a non-constant meromorphie function on the annulus A(R0) = [z : l/R0 < I zl < R}, where l < R < The function f is called a transcendental or

A( R )

T ( R, )

lim sup ——= œ, l < R < Ro = R^x log R

or

T ( R, )

lim sup —-—;—--- = oo, l < R < R0 < +oo,

R^Ro - log(Ro -R) c

respectively.

A

S (R, f) = o(T0 (R, f))

holds for all l < R < R except for the set Ar or the s et A'R mentioned in Theorem 1, respectively.

Next, we have

N (* J-;) =N {*• jh,) + N (R, )

=/1 ^^^^ ^+/R n2 ('; ^

in which each zero of the funetion f — a is eounted only once.

We use n!k) (t, j—^, respectively, n!(k (t, -j—^ j to denote the counting function of poles of the function with the multiplicities not exceeding k, respectively, greater than k in {z : t < |z| ^ 1}, where each point is counted only once. In a same way we introduce the notations

Wik\t, f), N(k(t, f), W2k\t, f), N2(k(t, f), W0k\t, f), N(k(t, f).

The following theorem was proved in [8].

Theorem 1. (The Second Fundamental Theorem on annuli). Let f be a non constant mero-morphic function on the annulus A = : < |z| < R^, where 1 < R < Ro ^ Let

ai, a2,..., a„ be q distinct complex numbers in the extended complex plane C = C U {^}, let k1,k2,..., kq be q positive integers, and let X ^ 0. Then

1 ^ v(i)/

(i) (q — 2)T0( R, f) < ^N0(r, j—;- ) — N(1\R, f) + 5 (R, f), (i i) (q — 2)To( R, f) < £ N0 (r, j—^j + 5 (R, f),

m , — 2)To(R n < ± ^ ^) (R, 7—-j) + ± ^ (r, )

(») (,— 2—£ ^) ro(R, /) < ± ^^) J—a-) + S(R, /),

N01)(R, f) = No (r, jj + 2No(R, f) — No(R, f)

+ 1 / f-0 kj + 1 V f — a

where

^0 ( R, J ) = No and it holds: 1. In the case Ro =

m^R,j) =0 (iog(RTo(R, /)))

for R E (1, except for the set AR such that

J Ra-1dR <

2. In the case Ro <

-K)=0( *( RRR

for R G (1, R0) except for the set AR such that

f dR

( Ro - R-i) <

A'

3. Main Results

Let f(z) be a meromorphic function on the annulus A = : < |z| < , where

1 < R < R0 ^ and a be a complex number in the extended complex plane C = C U We denote E(a, f) = {z G A : f(z) — a = 0}, where each zero with multiplicity m is counted m times. If we ignore the multiplicity, then the set is denoted by E(a, f). We use Ek)(a, f) to denote the set of zeros of f — a with multiplicities no greater than k, in which each zero is counted only once.

Our main result below is an analog of a result on the plane C obtained by H. S. Gopalkrishna and Subhas S. Bhoosnurmath [4].

Theorem 2. Let fi(z) and f2( z) be two transcendental meromorphic functions on the annulus A = : < |z| < , where 1 < R0 ^ Let aj (j = 1, 2,..., q) be q distinct complex numbers in C, and kj, j = 1, 2,... ,q be positive integers or satisfying

^ ... ^ kq. (1)

If _ _

Ek. )(aj ,h) = Ek. )(aj, f2), j = 1, 2,...,q, (2)

and

ijrU — ^rr >2- <3>

j=2 J

then fl(Z) = f2(z).

Proof. We assume that aj, j = 1, 2,..., q, are finite complex numbers, otherwise we make a suitable Mobius transformation. By Theorem 1 we have

to — 2^(R./■) < ¿^ ^1 {R- jhr,)

^^+TTo(R,/+ 5 (R,f i). =1

This implies

Therefore,

(t^ - 2) < 1 (R ^ + * <4>

kj +1 J j=; kj +1 V h — aj

Condition (1) implies:

fci k2 kq 1

1 > -— > -— > ... > -— > -

h + 1 k2 + 1 kq + 1 2

It follows from the above inequalities and (4) that

—2) t (R, /1) < ^ ± rt) (R, j—a-) + s (R, A). (5)

In the same way,

(t ¿T+T —2) T°(R- A) < ¡sr+T t) (R. ) + S(R- A). («>

Since fi(z) = f2(z), it follows from (2) that

max |g ) (R, ) , ¿; ) S N0(r, 1

j=1 . - - — aj) V /2 — aJ ) V J1 — j2

S ^ (R- 71^2 ) + 0(1)

S r„ (R, /1)+r„ (R, ¡2).

Therefore, from the above discussion we obtain

E ^ — 2j T0(R, /1) < [To(R, /1) + To(R, /2)] + S(R, h).

=1

Similarly,

j 1

(S ^—2)

k + 1 — 2 j To(R, /2) < [To(R, /1) + To(R, /2)] + S(R, /2).

Summing two above equations, we obtain:

E ^ — — 2 ) [To(R, /1) + To(R, /2)] < S(R, /1) + S(R, /2). (7)

=2

By (3) we get:

To(R, h) + To(R, /2) < S(R, /1) + S(R, /2), which is impossible since ^(z) and f2(z) are transcendental meromorphic functions. Hence, fi_(z) = f2(z). This completes the proof. □

From Theorem 2, we get the following corollary.

Corollary 1. Let fa(z) and f2(z) be two transcendental meromorphic functions on the an-nulus A = : ^ < |z| < Ro|, where 1 < R S Let aj, j = 1, 2,..., q, be q(^ 5) distinct complex numbers in C, and kj, j = 1, 2,..., q, be positive integers or satisfying

k1 ^ k2 ^^ kq.

and

Ekj )(aj, h) = Ekj )(aj, f2), j = 1 2,...,q.

Then

(i) ifq ^ 7, then f1(z) = f2(z).

(ii) if q = 6 and k3 ^ 2, then ^(z) = f2(z).

(iii) if q = 5, and k3 ^ 3 and k5 ^ 2, then f]_(z) = f2(z).

(iv) if q = 5 and k4 ^ 4, then f]_(z) = f2(z).

From Corollary 1, we obtain a following theorem.

Theorem 3. Let fi(z) and f2(z) be two transcendental meromorphic functions on the an-nulus A = : ^ < |z| < До I, where 1 < R0 ^ Let aj, j = 1,..., 7, be five distinct complex numbers in C. If E(aj, f\) = E(aj, f2) for j = 1,..., 7, then f\(z) = f2(z).

Corollary 1 implies the following analogue of Nevanlinna's five value theorem. In the case R0 = this statement was proved by Kondratyuk and Laine [31].

Theorem 4. Let fi(z) and f2(z) be two transcendental meromorphic functions on the an-nulus A = : ^ < |z| < , where 1 < R0 ^ Let aj, j = 1,..., 5, be five distinct complex numbers in C. If E(aj, f\) = E(aj, f2) for j = 1,..., 5, then f\(z) = f2(z).

The condition of fi(z) and f2(z) share five values in Theorem 3.3 can not be weakened to that fi(z) and f 2(z) share four values. For example, the functions fi(z) = ez and f 2(z) = e-z share four values 0, 1, -1, œ, but fi(z) = f 2( z).

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Ashok Rathod,

B.L.D.E.Association's

S.B. Arts and K.C.P. Science College,

Department of Mathematics,

SMT. Bangaramma Sajjan Campus,

Solapur Road, Vijayapura-586103,

Karnataka, India.

E-mail: ashokmrmaths@gmail.com