Articles
Kyungpook Mathematical Journal -0001; 56(3): 763-776
Published online November 30, -0001
Copyright © Kyungpook Mathematical Journal.
Uniqueness of Entire Functions that Share an Entire Function of Smaller Order with One of Their Linear Differential Polynomials
Xiao-Min Li1, Hong-Xun Yi2
Department of Mathematics, Ocean University Of China, Qingdao, Shandong 266100, P. R. China1
Department of Mathematics, Shandong University, Jinan, Shandong 250100, P. R. China2
Received: December 29, 2011; Accepted: March 21, 2014
Abstract
We prove a uniqueness theorem of entire functions sharing an entire function of smaller order with their linear differential polynomials. The results in this paper improve the corresponding results given by Gundersen-Yang[
Keywords: Entire functions, Shared values, Order of growth, Differential polynomials, Uniqueness theorems.
1. Introduction and main results
In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We adopt the standard notations in Nevanlinna theory of meromorphic functions as explained, e.g., in [5], [7] and [11]. It will be convenient to let
Let
Definition 1.1
For a nonconstant entire function
and
where and what follows,
In 1977, Rubel-Yang [8] proved that if an entire function
([4], Theorem 1)
Later on, Chang-Zhu [3] proved the following result to improve Theorem A:
([3], Theorem 1)
Consider the following linear differential polynomial related to
where and what follows,
We will prove the following result to improve Theorems A and B:
Theorem 1.2
L [f ]– a =c (f – a ), where c is some nonzero constant f is a solution of the equation L [f ]– a = (f – a )e p 1z +p 0such that σ (f ) =μ (f ) = 1, where not all a 0, a 1, · · ·,a k −1are zeros, p 1 ≠ 0and p 0are complex numbers.
From Theorem 1.2 we get the following corollary:
Corollary 1.3
Proceeding as in the proof of Theorem 1.2 in Section 3 of this paper, we get the following theorem:
Theorem 1.4
From Theorem 1.4 we get the following corollary:
Corollary 1.5
Example 1.6
Let
Example 1.7
([3]) Let
In 1995, Yi-Yang[12] posed the following question:
Question 1.8
([12], p.398) Let
Regarding Question 1.8, Gundersen-Yang [4] proved the following result:
([4], Theorem 2)
We will prove the following result to improve and complement Theorem E:
Theorem 1.9
f (z ) =γ 1e z , where γ 1 ≠ 0is a constant. f (z ) =γ 2e cz – [a (1– c )]/c, where n ≥ 2, γ 2 ≠ 0is a constant. L [f ] =f ′and f ′– a =c (f – a ), where c is some nonzero constant. f (z ) =γ 3e cz , where γ 3and c are two nonzero constants.
2. Preliminaries
In this section, we introduce some important results that will be used to prove the main results in this paper. First of all we introduce Wiman-Valiron theory. For this purpose, we first introduce the following notions: Let
Lemma 2.1
([7], Corollary 2.3.4)
Lemma 2.2
([6], Satz 4.5)
Lemma 2.3
(see [7], Lemma 1.1.2)
Lemma 2.4
([6], Satz 4.4)
Lemma 2.5
([9])
Lemma 2.6
([7], Remark of Corollary 2.3.5 or [12], Corollary of Theorem 1.21)
Lemma 2.7
([10], Theorem 1.1)
3. Proof of Theorems
Proof of Theorem 1.2
From the condition that
where
as
Suppose that
Then from (
From the condition that
as
where
as
and
Since
from (
as
as
where
as |
i.e.,
as
as
On the other hand, by Lemma 2.4 we know that
Since |
where
as
as
From (
From (
Suppose that
Then from (
We discuss the following two subcases.
Subcase 2.1
Suppose that
By (
Subcase 2.2
Suppose that
we will prove
Suppose that
Then from (
where
as
as
From (
By (
If
By Lemma 2.6 we know that
From (
and
From (
as
From (
which implies
From (
where
In fact, from (
Suppose that
From (
From (
From (
Since
From Definition 1.1, Lemma 2.3 Definition 1.1.1 and Theorem 1.1.3 from [13], and the assumption
as
as
Proof of Theorem 1.3
From Theorem B and the assumptions of Theorem 1.3 we know that there exists some nonzero constant
where
Suppose that there exists some nonzero constant
From (
where
If
From (
From (
Suppose that
From (
Suppose that
From (
By substituting (
If
Suppose that
such that
Suppose that
From (
If
If
From (
and so we have
From (
Acknowledgements
The authors wish to express their thanks to the referee for his/her valuable suggestions and comments.
References
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