Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 208516, 10 pages
doi:10.1155/2009/208516
Research Article

On Uniqueness of Meromorphic Functions with Multiple Values in Some Angular Domains

Zu-Xing Xuan1,2

1Department of Mathematical Sciences, Tsinghua University, Haidian District, Beijing 100084, China
2Department of Basic Courses, Beijing Union University, No. 97 Bei Si Huan Dong Road, Chaoyang District, Beijing 100101, China

Received 4 March 2009; Accepted 30 June 2009

Academic Editor: Narendra Kumar Govil

Copyright © 2009 Zu-Xing Xuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This article deals with problems of the uniqueness of transcendental meromorphic function with shared values in some angular domains dealing with the multiple values which improve a result of J. Zheng.

1. Introduction

A transcendental meromorphic function is meromorphic in the complex plane ℂ and not rational. We assume that the readers are familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as Nevanlinna deficiency 𝛿 ( 𝑎 , 𝑓 ) of 𝑓 ( 𝑧 ) with respect to ℂ 𝑎 ∈ and Nevanlinna characteristic 𝑇 ( 𝑟 , 𝑓 ) of 𝑓 ( 𝑧 ) . And the lower order 𝜇 and the order 𝜆 are in turn defined as follows: 𝜇 = 𝜇 ( 𝑓 ) = l i m i n f 𝑟 → ∞ l o g 𝑇 ( 𝑟 , 𝑓 ) , l o g 𝑟 𝜆 = 𝜆 ( 𝑓 ) = l i m s u p 𝑟 → ∞ l o g 𝑇 ( 𝑟 , 𝑓 ) . l o g 𝑟 ( 1 . 1 ) For the references, please see [1]. An 𝑎 ∈ ℂ = ℂ ∪ { ∞ } is called an IM (ignoring multiplicities) shared value in ℂ 𝑋 ⊆ of two meromorphic functions 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) if in 𝑋 , 𝑓 ( 𝑧 ) = 𝑎 if and only if 𝑔 ( 𝑧 ) = 𝑎 . It is Nevanlinna [2] who proved the first uniqueness theorem, called the Five Value Theorem, which says that two meromorphic functions 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) are identical if they have five distinct IM shared values in 𝑋 = ℂ . After his very fundamental work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigations (see [3]). Recently, Zheng in [4] suggested for the first time the investigation of uniqueness of a function meromorphic in a precise subset of ℂ , and this is an interesting topic.

Given 𝑚 pair of real numbers { 𝛼 𝑗 , 𝛽 𝑗 } satisfying − 𝜋 ≤ 𝛼 1 < 𝛽 1 ≤ 𝛼 2 < 𝛽 2 ≤ ⋯ ≤ 𝛼 𝑚 < 𝛽 𝑚 ≤ 𝜋 , ( 1 . 2 ) we define 𝜋 𝜔 = m a x 𝛽 1 − 𝛼 1 𝜋 , … , 𝛽 𝑚 − 𝛼 𝑚 . ( 1 . 3 )

Zheng in [4] proved the following theorem.

Theorem A Let 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) be both transcendental meromorphic functions, and let 𝑓 ( 𝑧 ) be of finite order 𝜆 and such that for some ℂ 𝑎 ∈ and an integer 𝑝 ≥ 0 , 𝛿 = 𝛿 ( 𝑎 , 𝑓 ( 𝑝 ) ) > 0 . For 𝑚 pair of real numbers { 𝛼 𝑗 , 𝛽 𝑗 } satisfying (1.2) and 𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 < 4 𝜎 a r c s i n 𝛿 2 , ( 1 . 4 ) where 𝜎 = m a x { 𝜔 , 𝜇 } , assume that 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) have five distinct IM shared values in ⋃ 𝑋 = 𝑚 𝑗 = 1 { 𝑧 ∶ 𝛼 𝑗 ≤ a r g 𝑧 ≤ 𝛽 𝑗 } . If 𝜔 < 𝜆 ( 𝑓 ) , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) .

However, it was not discussed whether there are similar results dealing with multiple values in some angular domains. In this paper we investigate this problem.

We use 𝐸 𝑘 ) ( 𝑎 , 𝑋 , 𝑓 ) to denote the set of zeros of 𝑓 ( 𝑧 ) − 𝑎 in 𝑋 , with multiplicities no greater than 𝑘 , in which each zero counted only once.

Our main result is what follows.

Theorem 1.1. Let 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) be both transcendental meromorphic functions, and let 𝑓 ( 𝑧 ) be of finite order 𝜆 and such that for some ℂ 𝑎 ∈ and an integer 𝑝 ≥ 0 , 𝛿 = 𝛿 ( 𝑎 , 𝑓 ( 𝑝 ) ) > 0 . For 𝑚 pair of real numbers { 𝛼 𝑗 , 𝛽 𝑗 } satisfying (1.2) and 𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 < 4 𝜎 a r c s i n 𝛿 2 , ( 1 . 5 ) where 𝜎 = m a x { 𝜔 , 𝜇 } , assume that 𝑎 𝑗 ( 𝑗 = 1 , 2 , … , 𝑞 ) are 𝑞 distinct complex numbers, and let 𝑘 𝑗 ( 𝑗 = 1 , 2 , … , 𝑞 ) be positive integers or ∞ satisfying 𝑘 1 ≥ 𝑘 2 ≥ ⋯ ≥ 𝑘 𝑞 , ( 1 . 6 ) 𝐸 𝑘 𝑗 ) 𝑎 𝑗 = , 𝑋 , 𝑓 𝐸 𝑘 𝑗 ) 𝑎 𝑗 , 𝑋 , 𝑔 , ( 1 . 7 ) 𝑞 𝑗 = 3 𝑘 𝑗 𝑘 𝑗 + 1 > 2 , ( 1 . 8 ) where ⋃ 𝑋 = 𝑞 𝑗 = 1 { 𝑧 ∶ 𝛼 𝑗 ≤ a r g 𝑧 ≤ 𝛽 𝑗 } . If 𝜔 < 𝜆 ( 𝑓 ) , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) .

2. Proof of Theorem 1.1

First we introduce several lemmas which are crucial in our proofs. The following result was proved in [5] (also see [6]).

Lemma 2.1 (see [5]). Let 𝑓 ( 𝑧 ) be transcendental and meromorphic in ℂ with the lower order 0 ≤ 𝜇 < ∞ and the order 0 < 𝜆 ≤ ∞ . Then for arbitrary positive number 𝜎 satisfying 𝜇 ≤ 𝜎 ≤ 𝜆 and a set 𝐸 with finite linear measure, there exists a sequence of positive numbers { 𝑟 𝑛 } such that (1) 𝑟 𝑛 ∈ 𝐸 , l i m 𝑛 → ∞ ( 𝑟 𝑛 / 𝑛 ) = ∞ ,(2) l i m i n f 𝑛 → ∞ ( l o g 𝑇 ( 𝑟 𝑛 , 𝑓 ) / l o g 𝑟 𝑛 ) ≥ 𝜎 , (3) 𝑇 ( 𝑡 , 𝑓 ) < ( 1 + 𝑜 ( 1 ) ) ( 𝑡 / 𝑟 𝑛 ) 𝜎 𝑇 ( 𝑟 𝑛 , 𝑓 ) , 𝑡 ∈ [ 𝑟 𝑛 / 𝑛 , 𝑛 𝑟 𝑛 ] .

A sequence 𝑟 𝑛 satisfying (1), (2), and (3) in Lemma 2.1 is called Polya peak of order 𝜎 outside 𝐸 in this article. For 𝑟 > 0 and 𝑎 ∈ ℂ define [ 𝐷 ( 𝑟 , 𝑎 ) ∶ = 𝜃 ∈ − 𝜋 , 𝜋 ) ∶ l o g + 1 | | 𝑓 𝑟 𝑒 𝑖 𝜃 | | > 1 − 𝑎 [ l o g 𝑟 𝑇 ( 𝑟 , 𝑓 ) , ( 2 . 1 ) 𝐷 ( 𝑟 , ∞ ) ∶ = 𝜃 ∈ − 𝜋 , 𝜋 ) ∶ l o g + | | 𝑓 𝑟 𝑒 𝑖 𝜃 | | > 1 l o g 𝑟 𝑇 ( 𝑟 , 𝑓 ) . ( 2 . 2 )

The following result is a special version of the main result of Baernstein [7].

Lemma 2.2. Let 𝑓 ( 𝑧 ) be transcendental and meromorphic in ℂ with the finite lower order 𝜇 and the order 0 < 𝜆 ≤ ∞ and for some 𝑎 ∈ ℂ , 𝛿 = 𝛿 ( 𝑎 , 𝑓 ) > 0 . Then for arbitrary Polya peak 𝑟 𝑛 of order 𝜎 > 0 , 𝜇 ≤ 𝜎 ≤ 𝜆 , we have l i m i n f 𝑛 → ∞ 𝑟 𝑚 𝑒 𝑠 𝐷 𝑛 4 , 𝑎 ≥ m i n 2 𝜋 , 𝜎 a r c s i n 𝛿 2 . ( 2 . 3 )

Although Lemma 2.2 was proved in [7] for the Polya peak of order 𝜇 , the same argument of Baernstein [7] can derive Lemma 2.2 for the Polya peak of order 𝜎 , 𝜇 ≤ 𝜎 ≤ 𝜆 .

Nevanlinna theory on angular domain will play a key role in the proof of theorems. Let 𝑓 ( 𝑧 ) be a meromorphic function on the angular domain Ω ( 𝛼 , 𝛽 ) = { 𝑧 ∶ 𝛼 ≤ a r g 𝑧 ≤ 𝛽 } , where 0 < 𝛽 − 𝛼 ≤ 2 𝜋 . Nevanlinna defined the following notations (see [8]): 𝐴 𝛼 , 𝛽 𝜔 ( 𝑟 , 𝑓 ) = 𝜋 𝑟 1 1 𝑡 𝜔 − 𝑡 𝜔 𝑟 2 𝜔 l o g + | | 𝑓 𝑡 𝑒 𝑖 𝛼 | | + l o g + | | 𝑓 𝑡 𝑒 𝑖 𝛽 | | 𝑑 𝑡 𝑡 , 𝐵 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) = 2 𝜔 𝜋 𝑟 𝜔 𝛽 𝛼 l o g + | | 𝑓 𝑟 𝑒 𝑖 𝜃 | | 𝐶 s i n 𝜔 ( 𝜃 − 𝛼 ) 𝑑 𝜃 , 𝛼 , 𝛽 = 2 | | 𝑏 1 < 𝑛 | | < 𝑟 1 | | 𝑏 𝑛 | | 𝜔 − | | 𝑏 𝑛 | | 𝜔 𝑟 2 𝜔 𝜃 s i n 𝜔 𝑛 , − 𝛼 ( 2 . 4 ) where 𝜔 = 𝜋 / ( 𝛽 − 𝛼 ) and 𝑏 𝑛 = | 𝑏 𝑛 | 𝑒 𝑖 𝜃 𝑛 are the poles of 𝑓 ( 𝑧 ) on Ω ( 𝛼 , 𝛽 ) appearing according to their multiplicities. 𝐶 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) is called the angular counting function of the poles of 𝑓 on Ω ( 𝛼 , 𝛽 ) and Nevanlinna's angular characteristic is defined as follows: 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) = 𝐴 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) + 𝐵 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) + 𝐶 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) . ( 2 . 5 )

Throughout, we denote by 𝑅 𝛼 , 𝛽 ( 𝑟 , ∗ ) a quantity satisfying 𝑅 𝛼 , 𝛽 ( 𝑟 , ∗ ) = 𝑂 l o g 𝑟 𝑆 𝛼 , 𝛽 ( 𝑟 , ∗ ) , 𝑟 ∈ 𝐸 , ( 2 . 6 ) where 𝐸 denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context [9].

Lemma 2.3. Let 𝑓 ( 𝑧 ) be meromorphic on Ω ( 𝛼 , 𝛽 ) . Then for arbitrary complex number 𝑎 , we have 𝑆 𝛼 , 𝛽 1 𝑓 − 𝑎 = 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) + 𝑂 ( 1 ) , ( 2 . 7 ) and for an integer 𝑝 ≥ 0 , 𝑆 𝛼 , 𝛽 𝑟 , 𝑓 ( 𝑝 ) ≤ 2 𝑝 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) + 𝑅 𝛼 , 𝛽 𝐴 ( 𝑟 , 𝑓 ) , 𝛼 , 𝛽 𝑓 𝑟 , ( 𝑝 ) 𝑓 + 𝐵 𝛼 , 𝛽 𝑓 𝑟 , ( 𝑝 ) 𝑓 = 𝑅 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) , ( 2 . 8 ) and 𝑅 𝛼 , 𝛽 ( 𝑟 , 𝑓 ( 𝑝 ) ) = 𝑅 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) .

Lemma 2.4. Let 𝑓 ( 𝑧 ) be meromorphic on Ω ( 𝛼 , 𝛽 ) . Then for arbitrary 𝑞 distinct 𝑎 𝑗 ∈ ℂ ( 1 ≤ 𝑗 ≤ 𝑞 ) , we have ( 𝑞 − 2 ) 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) ≤ 𝑞 𝑗 = 1 𝐶 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑅 ( 𝑟 , 𝑓 ) , ( 2 . 9 ) where the term 𝐶 𝛼 , 𝛽 ( 𝑟 , 1 / ( 𝑓 − 𝑎 𝑗 ) ) will be replaced by 𝐶 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) when some 𝑎 𝑗 = ∞ .

We use 𝐶 𝑘 ) 𝛼 , 𝛽 ( 𝑟 , 1 / ( 𝑓 − 𝑎 ) ) to denote the zeros of 𝑓 ( 𝑧 ) − 𝑎 in Ω ( 𝛼 , 𝛽 ) whose multiplicities are no greater than 𝑘 and are counted only once. Likewise, we use 𝐶 ( 𝑘 + 1 𝛼 , 𝛽 ( 𝑟 , 1 / ( 𝑓 − 𝑎 ) ) to denote the zeros of 𝑓 ( 𝑧 ) − 𝑎 in Ω ( 𝛼 , 𝛽 ) whose multiplicities are greater than 𝑘 and are counted only once.

Lemma 2.5. Let 𝑓 ( 𝑧 ) be meromorphic on Ω ( 𝛼 , 𝛽 ) , and let 𝑘 𝑗 ( 𝑗 = 1 , 2 , … , 𝑞 ) be 𝑞 positive integers. Then for arbitrary 𝑞 distinct 𝑎 𝑗 ∈ ℂ ( 1 ≤ 𝑗 ≤ 𝑞 ) , we have ( i ) ( 𝑞 − 2 ) 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) < 𝑞 𝑗 = 1 𝑘 𝑗 𝑘 𝑗 + 1 𝐶 𝑘 𝑗 ) 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑞 𝑗 = 1 1 𝑘 𝑗 𝐶 + 1 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑅 ( 𝑟 , 𝑓 ) , ( i i ) 𝑞 − 2 − 𝑞 𝑗 = 1 1 𝑘 𝑗 𝑆 + 1 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) < 𝑞 𝑗 = 1 𝑘 𝑗 𝑘 𝑗 + 1 𝐶 𝑘 𝑗 ) 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑅 ( 𝑟 , 𝑓 ) , ( 2 . 1 0 ) where the term 𝐶 𝛼 , 𝛽 ( 𝑟 , 1 / ( 𝑓 − 𝑎 𝑗 ) ) will be replaced by 𝐶 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) when some 𝑎 𝑗 = ∞ .

Proof. According to our notations, we have 𝐶 𝛼 , 𝛽 1 𝑟 , = 𝑓 − 𝑎 𝐶 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 𝑓 − 𝑎 𝐶 ( 𝑘 + 1 𝛼 , 𝛽 1 𝑟 , = 𝑘 𝑓 − 𝑎 𝑘 + 1 𝐶 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 1 𝑓 − 𝑎 𝑘 + 1 𝐶 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 𝑓 − 𝑎 𝐶 ( 𝑘 + 1 𝛼 , 𝛽 1 𝑟 , ≤ 𝑘 𝑓 − 𝑎 𝑘 + 1 𝐶 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 1 𝑓 − 𝑎 𝐶 𝑘 + 1 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 1 𝑓 − 𝑎 𝐶 𝑘 + 1 ( 𝑘 + 1 𝛼 , 𝛽 1 𝑟 , = 𝑘 𝑓 − 𝑎 𝑘 + 1 𝐶 𝑘 ) 𝛼 , 𝛽 1 𝑟 , + 1 𝑓 − 𝑎 𝐶 𝑘 + 1 𝛼 , 𝛽 1 𝑟 , . 𝑓 − 𝑎 ( 2 . 1 1 ) By Lemma 2.4, ( 𝑞 − 2 ) 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) ≤ 𝑞 𝑗 = 1 𝐶 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 ≤ + 𝑅 ( 𝑟 , 𝑓 ) 𝑞 𝑗 = 1 𝑘 𝑗 𝑘 𝑗 + 1 𝐶 𝑘 𝑗 ) 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑞 𝑗 = 1 1 𝑘 𝑗 𝐶 + 1 𝛼 , 𝛽 1 𝑟 , 𝑓 − 𝑎 𝑗 + 𝑅 ( 𝑟 , 𝑓 ) , ( 2 . 1 2 ) and (i) follows.
Furthermore, 𝐶 𝛼 , 𝛽 ( 𝑟 , 1 / ( 𝑓 − 𝑎 𝑗 ) ) < 𝑆 𝛼 , 𝛽 ( 𝑟 , 𝑓 ) , and on combining this with (i), we get (ii).

Proof of Theorem 1.1. Suppose 𝑓 ( 𝑧 ) ≢ 𝑔 ( 𝑧 ) . For convenience, below we omit the subscript of all the notations, such as 𝑆 ( 𝑟 , ∗ ) and 𝐶 ( 𝑟 , ∗ ) . By applying Lemma 2.5 to 𝑔 and (1.6), we have 𝑞 𝑗 = 3 𝑘 𝑗 𝑘 𝑗 + + 1 2 𝑘 2 𝑘 2 𝑘 + 1 − 2 𝑆 ( 𝑟 , 𝑔 ) ≤ 2 𝑘 2 + 1 𝑞 𝑗 = 1 𝐶 𝑘 𝑗 ) 1 𝑟 , 𝑔 − 𝑎 𝑗 ≤ 𝑘 + 𝑅 ( 𝑟 , 𝑔 ) 2 𝑘 2 𝐶 1 + 1 𝑟 , ≤ 𝑘 𝑓 − 𝑔 + 𝑅 ( 𝑟 , 𝑔 ) 2 𝑘 2 ≤ 𝑘 + 1 𝑆 ( 𝑟 , 𝑓 − 𝑔 ) + 𝑅 ( 𝑟 , 𝑔 ) 2 𝑘 2 𝑘 + 1 𝑆 ( 𝑟 , 𝑓 ) + 2 𝑘 2 + 1 𝑆 ( 𝑟 , 𝑔 ) + 𝑅 ( 𝑟 , 𝑔 ) , ( 2 . 1 3 ) so that 𝑞 𝑗 = 3 𝑘 𝑗 𝑘 𝑗 + 𝑘 + 1 2 𝑘 2 𝑘 + 1 − 2 𝑆 ( 𝑟 , 𝑔 ) − 𝑅 ( 𝑟 , 𝑔 ) < 2 𝑘 2 + 1 𝑆 ( 𝑟 , 𝑓 ) . ( 2 . 1 4 ) This implies that 𝑅 ( 𝑟 , 𝑔 ) = 𝑅 ( 𝑟 , 𝑓 ) . We have also (2.14) for alternation of 𝑓 and 𝑔 , then 𝑞 𝑗 = 3 𝑘 𝑗 𝑘 𝑗 + 𝑘 + 1 2 𝑘 2 𝑘 + 1 − 2 𝑆 ( 𝑟 , 𝑓 ) − 𝑅 ( 𝑟 , 𝑓 ) < 2 𝑘 2 + 1 𝑆 ( 𝑟 , 𝑔 ) ≤ 𝑆 ( 𝑟 , 𝑓 ) + 𝑅 ( 𝑟 , 𝑓 ) . ( 2 . 1 5 ) By (1.8), we have 𝑆 ( 𝑟 , 𝑓 ) = 𝑂 ( l o g 𝑟 ) , 𝑟 ∉ 𝐸 . ( 2 . 1 6 )
We assume that 𝑎 ∈ ℂ . By the same argument we can show Theorem 1.1 for the case when 𝑎 = ∞ . By applying Lemma 2.3 and (2.16), we estimate
𝐵 1 𝑟 , 𝑓 ( 𝑝 ) − 𝑎 ≤ 𝑆 𝑟 , 𝑓 ( 𝑝 ) = 𝑓 + 𝑂 ( 1 ) ( 𝐴 + 𝐵 ) 𝑟 , ( 𝑝 ) 𝑓 + ( 𝐴 + 𝐵 ) ( 𝑟 , 𝑓 ) + 𝑝 𝐶 ( 𝑟 , 𝑓 ) + 𝐶 ( 𝑟 , 𝑓 ) + 𝑂 ( 1 ) ≤ ( 𝑝 + 1 ) 𝑆 ( 𝑟 , 𝑓 ) + 𝑅 ( 𝑟 , 𝑓 ) = 𝑂 ( l o g 𝑟 ) , 𝑟 ∉ 𝐸 . ( 2 . 1 7 )
The following method comes from [10]. But we quote it in detail here because of its independent significance. Note that 𝜆 ( 𝑓 ) > 𝜔 . We need to treat two cases.
(I)   𝜆 ( 𝑓 ) > 𝜇 . Then 𝜆 ( 𝑓 ( 𝑝 ) ) = 𝜆 ( 𝑓 ) > 𝜎 ≥ 𝜇 = 𝜇 ( 𝑓 ( 𝑝 ) ) . And by the inequality (1.5), we can take a real number 𝜖 > 0 such that
𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 4 + 2 𝜖 + 2 𝜖 < 𝜎 + 2 𝜖 a r c s i n 𝛿 2 , ( 2 . 1 8 ) where 𝛼 𝑚 + 1 = 2 𝜋 + 𝛼 1 , and 𝜆 𝑓 ( 𝑝 ) > 𝜎 + 2 𝜖 > 𝜇 . ( 2 . 1 9 ) Applying Lemma 2.1 to 𝑓 ( 𝑝 ) ( 𝑧 ) gives the existence of the Polya peak 𝑟 𝑛 of order 𝜎 + 2 𝜖 of 𝑓 ( 𝑝 ) such that 𝑟 𝑛 ∉ 𝐸 , and then from Lemma 2.2 for sufficiently large 𝑛 we have 𝑟 𝑚 𝑒 𝑠 𝐷 𝑛 > 4 , 𝑎 𝜎 + 2 𝜖 a r c s i n 𝛿 2 − 𝜖 , ( 2 . 2 0 ) since 𝜎 + 2 𝜖 > 1 / 2 . We can assume for all the 𝑛 , (13) holds. Set 𝐷 𝑟 𝐾 ∶ = 𝑚 𝑒 𝑠 𝑛 , 𝑎 𝑚 𝑗 = 1 𝛼 𝑗 + 𝜖 , 𝛽 𝑗 − 𝜖 . ( 2 . 2 1 ) Then from (2.18) and (2.20) it follows that 𝐷 𝑟 𝐾 ≥ 𝑚 𝑒 𝑠 𝑛 [ , 𝑎 − 𝑚 𝑒 𝑠 0 , 2 𝜋 ) ⧵ 𝑚 𝑗 = 1 𝛼 𝑗 + 𝜖 , 𝛽 𝑗 𝐷 𝑟 − 𝜖 = 𝑚 𝑒 𝑠 𝑛 , 𝑎 − 𝑚 𝑒 𝑠 𝑚 𝑗 = 1 𝛽 𝑗 − 𝜖 , 𝛼 𝑗 + 1 𝐷 𝑟 + 𝜖 = 𝑚 𝑒 𝑠 𝑛 − , 𝑎 𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 + 2 𝜖 > 𝜖 > 0 . ( 2 . 2 2 ) It is easy to see that there exists a 𝑗 0 such that for infinitely many 𝑛 , we have 𝐷 𝑟 𝑚 𝑒 𝑠 𝑛 𝛼 , 𝑎 𝑗 0 + 𝜖 , 𝛽 𝑗 0 > 𝐾 − 𝜖 𝑞 . ( 2 . 2 3 ) We can assume for all the 𝑛 , (2.23) holds. Set 𝐸 𝑛 = 𝐷 ( 𝑟 𝑛 ⋂ , 𝑎 ) ( 𝛼 − 𝑗 0 + 𝜖 , 𝛽 𝑗 0 − 𝜖 ) . Thus from the definition (2.1) of 𝐷 ( 𝑟 , 𝑎 ) it follows that 𝛽 𝑗 0 𝛼 − 𝜖 𝑗 0 + 𝜖 l o g + 1 | | 𝑓 ( 𝑝 ) 𝑟 𝑛 𝑒 𝑖 𝜃 | | − 𝑎 𝑑 𝜃 ≥ 𝐸 𝑛 l o g + 1 | | 𝑓 ( 𝑝 ) 𝑟 𝑛 𝑒 𝑖 𝜃 | | 𝐸 − 𝑎 𝑑 𝜃 ≥ 𝑚 𝑒 𝑠 𝑛 𝑇 𝑟 𝑛 , 𝑓 ( 𝑝 ) l o g 𝑟 𝑛 > 𝐾 𝑚 𝑇 𝑟 𝑛 , 𝑓 ( 𝑝 ) l o g 𝑟 𝑛 . ( 2 . 2 4 )
On the other hand, by the definition (2.4) of 𝐵 𝛼 , 𝛽 ( 𝑟 , ∗ ) and (2.14), we have
𝛽 𝑗 0 𝛼 − 𝜖 𝑗 0 + 𝜖 l o g + 1 | | 𝑓 ( 𝑝 ) 𝑟 𝑛 𝑒 𝑖 𝜃 | | 𝜋 − 𝑎 𝑑 𝜃 ≤ 2 𝜔 𝑗 0 s i n 𝜖 𝜔 𝑗 0 𝑟 𝜔 𝑗 0 𝐵 𝛼 𝑗 0 , 𝛽 𝑗 0 1 𝑟 , 𝑓 ( 𝑝 ) < 𝐾 − 𝑎 𝑗 0 𝑟 𝜔 𝑗 0 l o g 𝑟 , 𝑟 ∉ 𝐸 . ( 2 . 2 5 ) Combining (2.24) with (2.25) gives 𝑇 𝑟 𝑛 , 𝑓 ( 𝑝 ) ≤ 𝑚 𝐾 𝑗 0 𝐾 𝑟 𝜔 𝑗 0 𝑛 l o g 2 𝑟 𝑛 . ( 2 . 2 6 ) Thus from (1.5) in Lemma 2.1 for 𝜎 + 2 𝜖 , we have 𝜎 + 𝜖 ≤ l i m s u p 𝑛 → ∞ 𝑟 l o g 𝑇 𝑛 , 𝑓 ( 𝑝 ) l o g 𝑟 𝑛 ≤ 𝜔 𝑗 0 ≤ 𝜎 + 𝜖 . ( 2 . 2 7 ) This is impossible.
(II)   𝜆 ( 𝑓 ) = 𝜇 . Then 𝜎 = 𝜇 = 𝜆 ( 𝑓 ) = 𝜆 ( 𝑓 ( 𝑝 ) ) = 𝜇 ( 𝑓 ( 𝑝 ) ) . By the same argument as in (I) with all the 𝜎 + 2 𝜖 replaced by 𝜎 = 𝜇 , we can derive
m a x { 𝜔 , 𝜇 } = 𝜎 ≤ 𝜔 < 𝜆 ( 𝑓 ) . ( 2 . 2 8 ) This is impossible. Theorem 1.1 follows.

Remark 2.6. In Theorem A, 𝑞 = 5 , 𝑘 1 = 𝑘 2 = 𝑘 3 = 𝑘 4 = 𝑘 5 = ∞ , then 𝑘 3 𝑘 3 + 𝑘 + 1 4 𝑘 4 + 𝑘 + 1 5 𝑘 5 + 1 = 3 > 2 , ( 2 . 2 9 ) so Theorem A is a special case of Theorem 1.1. Meanwhile, Zheng in [4, pages 153–154] gave some examples to indicate that the conditions are necessary. So the conditions in theorem are also necessary.

Corollary 2.7. In Theorem 1.1, (i)if 𝑞 = 7 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,(ii)if 𝑞 = 6 , 𝑘 3 ≥ 2 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,(iii)if 𝑞 = 5 , 𝑘 3 ≥ 3 , 𝑘 5 ≥ 2 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,(iv)if 𝑞 = 5 , 𝑘 4 ≥ 4 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,(v)if 𝑞 = 5 , 𝑘 3 ≥ 5 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,(vi)if 𝑞 = 5 , 𝑘 3 ≥ 6 , 𝑘 4 ≥ 2 , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) ,

Corollary 2.8. Let 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) be both transcendental meromorphic functions and let 𝑓 ( 𝑧 ) be of finite lower order 𝜇 and such that for some ℂ 𝑎 ∈ and an integer 𝑝 ≥ 0 , 𝛿 = 𝛿 ( 𝑎 , 𝑓 ( 𝑝 ) ) > 0 . For 𝑚 pair of real numbers { 𝛼 𝑗 , 𝛽 𝑗 } satisfying (1.2) and 𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 < 4 𝜎 a r c s i n 𝛿 2 , ( 2 . 3 0 ) where 𝜎 = m a x { 𝜔 , 𝜇 } , assume that 𝑎 𝑗 ( 𝑗 = 1 , 2 , … , 𝑞 ) are 𝑞 ( = 5 + [ 2 / 𝑘 ] ) distinct complex numbers satisfying that 𝐸 𝑘 ) ( 𝑎 𝑗 , 𝑋 , 𝑓 ) = 𝐸 𝑘 ) ( 𝑎 𝑗 , 𝑋 , 𝑔 ) ( 𝑗 = 1 , 2 , … , 𝑞 ) , where 𝑘 is an integer or ∞ . If 𝜔 < 𝜆 ( 𝑓 ) , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) .

Corollary 2.9. Let 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) be both transcendental meromorphic functions and let 𝑓 ( 𝑧 ) be of finite lower order 𝜇 and such that for some ℂ 𝑎 ∈ and an integer 𝑝 ≥ 0 , 𝛿 = 𝛿 ( 𝑎 , 𝑓 ( 𝑝 ) ) > 0 . For 𝑚 pair of real numbers { 𝛼 𝑗 , 𝛽 𝑗 } satisfying (1.2) and 𝑚 𝑗 = 1 𝛼 𝑗 + 1 − 𝛽 𝑗 < 4 𝜎 a r c s i n 𝛿 2 , ( 2 . 3 1 ) where 𝜎 = m a x { 𝜔 , 𝜇 } , assume that 𝑎 𝑗 ( 𝑗 = 1 , 2 , … , 𝑞 ) are 𝑞 = 5 distinct complex numbers satisfying that 𝐸 3 ) ( 𝑎 𝑗 , 𝑋 , 𝑓 ) = 𝐸 3 ) ( 𝑎 𝑗 , 𝑋 , 𝑔 ) ( 𝑗 = 1 , 2 , 3 ) , 𝐸 2 ) ( 𝑎 𝑗 , 𝑋 , 𝑓 ) = 𝐸 2 ) ( 𝑎 𝑗 , 𝑋 , 𝑔 ) ( 𝑗 = 4 , 5 ) , then 𝑓 ( 𝑧 ) ≡ 𝑔 ( 𝑧 ) .

Question 1. For two meromorphic functions defined in ℂ , there are many uniqueness theorems when they share small functions ( 𝑎 ( 𝑧 ) is called a small function of 𝑓 ( 𝑧 ) if 𝑇 ( 𝑟 , 𝑎 ( 𝑧 ) ) = 𝑜 ( 𝑇 ( 𝑟 , 𝑓 ) ) ( 𝑟 → ∞ ) ) (see [3]). So we ask an interesting question: are there similar results when they share small functions in some precise domain 𝑋 ⊆ ℂ ?

Acknowlegment

The work is supported by NSF of China (no. 10871108).

References

  1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
  2. R. Nevanlinna, Le théorème de Picard-Borel et la Théorie des Fonctions Méromorphes, Gauthier-Villars, Paris, France, 1929.
  3. H. X. Yi and C.-C. Yang, Uniqueness Theorey of Meromorphic Functions, Kluwer Academic Publishers, Boston, Mass, USA, 2003.
  4. J. H. Zheng, “On uniqueness of meromorphic functions with shared values in some angular domains,” Canadian Mathematical Bulletin, vol. 47, pp. 152–160, 2004.
  5. L. Yang, “Borel directions of meromorphic functions in an angular domain,” Sci.Sinica, pp. 149–163, 1979.
  6. A. Edrei, “Meromorphic functions with three radially distributed values,” Transactions of the American Mathematical Society, vol. 78, pp. 276–293, 1955.
  7. A. Baernstein, “Proof of Edrei's spread conjecture,” Proceedings of the London Mathematical Society, vol. 26, pp. 418–434, 1973.
  8. A. A. Goldberg and I. V. Ostrovskii, The Distribution of Values of Meromorphic Functions, Nauka, Moscow, Russia, 1970.
  9. L. Yang and C.-C. Yang, “Angular distribution of value of ff′,” Science in China, vol. 37, pp. 284–294, 1994.
  10. J. H. Zheng, “On transcendental meromorphic functions with radially distributed values,” Science in China, vol. 47, pp. 401–416, 2004.