Nevanlinna’s five-value theorem for algebroid functions Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 10. № 2 (2018). С. 127-132.

УДК 512.5

NEVANLINNA'S FIVE-VALUE THEOREM FOR ALGEBROID FUNCTIONS

ASHOK RATHOD

Abstract. By using the second main theorem of the algebroid function, we study the following problem. Let W1(z) and W2(z) be two ^-valued non-constant algebroid functions, aj (j = 1, 2,... ,q) be q ^ 4u + 1 distinct complex numbers or to. Suppose that k1 ^ k2 ^ ... ^ kq, m are positive integers or to, 1 ^ m ^ q and 5j ^ 0 j = 1,2,... ,q, are such that

(l + tM V + + V 5j < (q - m - 1) (1 + + m.

V w^ 1 + kj j= V w

Let Bj = Ekj (aj ,f)\Ek. (aj ,5) for j = 1, 2,...,q. If

w 1 — aj ^

and

N*< ) < «j т мп)

ч _ -,

£ Nk] (г, ^) vk lim inf —- >

r^oo q __, q k q

£ Nk, (r, w—r) (1 + km)J2 kk+1 - 2y(1 + km) + (rn - 2u - £ 5j)km j=i 2 J j=i J j=i

then W1(z) = W2(z). This result improves and generalizes the previous results given by

Xuan and Gao.

Keywords: value distribution theory, Nevanlinna theory, algebroid functions, uniqueness.

Subject Classification: 30D35

1. Introduction

The value distribution theory of meromorphie functions was extended to the corresponding theory of algebroid functions by Ullarieh [1] and Valiron [2] around 1930, and important results on uniqueness for algebroid functions were obtained. It is well known that Valiron obtained a famous (4u + 1)-valued theorem. The uniqueness theory of algebroid functions is an interesting problem in the value distribution theory. Many researchers like Valiron [2], Baganas [3], He [4] and others ([6],[7],[9-27]) made lot of work in this area. In this article, we extend a result by Indrajit Lahiri and Eupa Pal [5] in the Nevanlinna's value distribution theory of meromorphie functions on Nevanlinna's five values theorem to algebroid functions

Let Av (z),Au-1(z),... ,A0(z) be analytic functions with no common zeros in the complex plane and consider the equation

A, (z )WV + Av_i{z)Wv~1 + ... + Ai(z)W + Aa(z) = 0. (1)

Ashok Rathod, Nevanlinna's five-value theorem for algebroid functions. ©Ashok Rathod 2018 .

The author is supported by the UGC-Rajiv Gandhi National Fellowship (no. F1-17.1/2013-14-SC-KAR-40380) of India.

Поступила 06 апреля 2017 г.

This equation defines a //-valued algebroid function W(z) [8],

It is well known [8] that on the complex plane with the projection of the critical points of the function W cut out, the Nevanlinna characteristic T(r, W) is defined as

T(r, W) = m(r, W) + N(r, W),

where

i v r-2n

m(r, W) =-V log+ Iwj(reid)ldd,

2tTv f^Jo

Ar. 1 [r n(t,W) -n(0,W) , n(0,W)n

N (r,W ) = - ——--——- dt + — log r.

o

Let Wi(z) and mj(z) be one-valued branches of two (^-valued and //-valued) algebroid functions, Following Prokopovich [15], we consider their quotient in the domain of the complex plane with the projection of the critical points of both functions cut out. The one-valued branches of th function W/M (W • M) are defined as Wi/mj (wi • mj), where 1 ^ i ^ m, 1 ^ j ^ n. The Nevanlinna's characteristic T(r, W/M) is defined bv T(r, W) + T(r, M),

Lemma 1 (8). Let W(z) be a v-valued algebroid function and {aj Yj=1 C C be q distinct complex numbers and let {kj }Qj=1 C N be q positive integers. Then

q k — q 1 (q - 2V)T (r, W) ^ Nk]) (r, W = a,) + J] -j—N(r, W = a,) + S(r, W),

k=1 j k=1 j

(-*-è™sèk -

Nk]) (r, W = aj) + N(r, W = aj) + S(r, W).

= k + 1) £ik + 1 In 2006 Zu-Xing Xuan and Zong-Sheng Gao [18] improved this statement as follows.

Theorem 1. Let W(z) and M(z) be two u-valued non-constant algebroid functions, let aj (j = 1, 2,..., 4u + 1) be + 1 distinct complex nu,mbers in C. If

Ë2u+i)(a3, W ) = Ë2u+1)(a3, M ), j = 1, 2,..., 2u + 1

and

Ë2v)(aj, W) = Ë2v)(aj, M), j = 1, 2,..., 4u+1,

W( ) = M( )

2. Main Results

Let W(z) be a //-valued algebroid function and a e C be a complex number. The symbol Ek (W = a) denotes the set of zeros of W(z) — a, whose multiplicities are not greater than k. The symbol nk)(W = a) stands for the number of distinct zeros of W(z) — a in |z| ^ r, whose multiplicities do not exceed k and are counted only once. Similarly, we define the functions u[k+i(r,W = a), Nk)(r,W = a) and N{k+i(r,W = a).

In this paper, we study the problem on the Nevanlinna's five value theorem for algebroid functions. To state our main theorem, we first introduce the following definition.

Definition 1. For B C A and a e C, we denote by NB (r, ) the reduced counting function of the zeros of f — a on A belonging to the set B.

Theorem 2. Let W1(z) rnd W2(z) be two v-valued non-constant algebroid functions, let aj (j = 1,2,..., q) be q ^ 4u + 1 distinct complex numb ers or x>. Suppose that

ki ^ k2 ^ ... ^ kq,m are positive integers or 1 ^ m ^ q and öj ^ 0, j = 1, 2, are such that

0+km,) t rrij+3"+t * <«-m -10+km,)

N ' j=m J j=i N '

Let Bj = Ekj (aj, f)\Ek. (aj, g) for j = 1, 2-...- q. If

NBj(r,w-1--) ^ 5jT(r,Wl)

(2)

(3)

and

lim inf

=1

ENkj (r, Wi-j)

v km

q -->--

E Nkj (T, W—aTj ) (1 + km) E kj+1 - M1 + km =1 =1

(4)

m - 2 - j) km =1

then m(z) = W2(z).

Proof Suppose that W\(z) = (z). The by Lemma 1 for each integer m, 1 ^ m ^ q, we have

1

(q - 2 u) T (r, W) ^YN ir,—^-^ + S (r, W)

j=l V W - ajJ

1

W - j 1

(^W^)} + S <r )

=1 =1

^y \Nkj)( ^ =1

< t^N.)(n^^U-V^r 1

q r £ {Nk =1

q f

5 iNk

=1

1 + k

W - a^ 1 + kj

kj)

+ -r-^N{k j+Jr 1

ir,W1-aJ)}

(r,-L_ ) +--1—N U^- H

V ,W1 - aj) 1 + kj V,W - aj)]

+ S( , W )

+ S( , W )

^ E Y+kh^Nkj)(+ Y.T^rT (r , W) + S (r, W)

j / . , 1 + kj =1

{r-W^T) + £ t (j+hj- T+mm,)Nkj) (-w^) + ()

W - aj J ' 1 + kj

=

+ £ Nj){ r>wh, I+S (r '

« £ =1

k / 1

1 + km

Nkj) I r

{r - Wi - aj) ( )

+ m - 1 -

( m - 1) km

m

+E =1

kj \ 1 + kj)

T( , W ) + S( , W )

Therefore

(y k

V ^ kj + 1

= m

- 2 +

( m - 1) m

m

T (r-W) ^

k

<vm. "TT

=1

Nkj)[ r

( r- W - aj)

+ S (r-W). (5)

k

m

Similarly,

(y ki

{^k, + 1 \j=m J

k„

k, — 2u+(m — 1)k^T(rW) < ± km - f 1

=1

^-tN,i^W—a)+S^ <«>

Since Bj = Ekj )(aj ,Wi) Ekj )(aj, W2), let Dj = Ekj )(aj ,Wi) B,- for j = 1, 2,...,q. Thus, by (5) and (6), for a sequence of values of r tending to we get:

YWk.)( r,-1-^ =y NB.)( r,-1-Vy Nd. )( r,-1-^

Z^ H'Wi — aJ^ B] H'Wi — ajj H'Wi — aj

j=m N J / j=m x J / J=m N J /

( Avj + r^fr.W—W;)

j=m ^ '

^ (u+ ^ M T(r, Wi) + vT(r, W2) + 0(1)

= m

Therefore,

Since

= m

—2 +

(m — 1)km

km

+ 0W)N^)( )

± -¿Ui -W—^)

= m =1

+ < -+ 0«) ± jt+1N*) {'-w^ )■

1 >

h

> ... >

k„

1

^ -,

kx + 1 kq + 1 2'

by (7), for a sequence of values of r tending to we get

= m

(7)

kj _2ij+(m — 1)k-

m m

m

m

km

{'++m)N")(a;)

i ( u + 0(l))

m

q — { ) ^

=1

1

W2 — a,

This implies that

r . ,) [r, wh;

limini =-^-

Nh ) r _—

Nkj1 I 1 , W2 _a

(

E

= m

kj+i

—2 +

m — 2 —

= m

V km

I ^ kj

I kj+i

\3=m

(1 + km) E T+ — 2v(1 + km) + km\m — 2v

m

= m

This i contradicts equation (2), Thus, we have f(z) ^ g(z). The proof is complete.

□

Theorem 2 yield the following corollaries.

k

k

m

k

Corollary 1. Let m = 1, kj = œ for j = 1, 2, 3,... ,q and

Ж ^ 1

kj) [r, Wi-aJ 1

7 = lim iní-p-^ > -

™ N, л (r^^^) V — 2u +1

kj) y, W2-aj

If NB. (r, Wl—a. ) ^ öjT(r, W\), where 8 ^ 0 satisfies

l

0 ^ Y15i <k — (2U + 1) — ",

3=i 7

then f (z) = g(z)

If we take q = E(aj, f) С E(aj, f), then Bj = 0 for j = 1, 2,..., +1. Therefore,

if we choose 8j = 0 for j = 1, 2,..., + 1 and take any constant 7 obeying 0 ^ 2u — ^ in

Corollary 1, we can get that f = g. Moreover, if q = E(aj, f) = E(aj, g), then 7 = 1

and 8j = 0 for j = 1, 2,..., + 1; this implies f = g. Then Corollary 1 is an improvement of Theorem 1,

Corollary 2. Let W\(z) and W2(z) be two v-valued non-constant algebroid functions, let aj, j = 1, 2,... ,q, be q ^ 5 distinct complex питbers or <x. Suppose that k\,k2,... ,k— are positive integers or with k\ ^ k2 ^ ... ^ kg if Ek.)(aj, f) С Ek.)(aj,g) and

v—л kj k\

£ ЙТГ1 — ЖГГ) — *> 0

where 7 is as stated in Corollary 1. Then f (z) = g(z).

Corollary 3. Under the assumptions of Corollary 2, we have Ek.)(aj,W\) = Ek.)(aj,W2) and

v—л kj k\

£ ЙТГ1 — WTT)— *> 0

Corollary 4. Let W\(z) and W2(z) be two v-valued non-constant algebroid functions, let aj, j = 1, 2,... ,q, be q ^ 5 distinct complex nu,m bers or <x. Suppose that k\,k2,... ,k— are positive integers or with k\ ^ k2 ^ ... ^ k— if Ek.)(aj, f) С Ek.)(aj,g) and

JL ¡?. (m — 2u — 1 )km

У —~— — 2p + ----^ — 2P > 0,

k3 + 1 -i(km + 1) '

where 7 is as stated a in Corollary 1. Then f (z) = g(z).

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Ashok Rathod,

Department of Mathematics,

Karnatak University,

Dharwad-580003,India.

E-mail: ashokmrmaths@gmail.com