Value sharing results for shifts of meromorphic functions

Abstract

In this paper, we deal with the value distribution of difference products of entire functions, and present some result on two difference products of entire functions sharing one value with the same multiplicities. The research findings also include an analogue for shift of a well-known conjecture by Brück. Our theorems improve the results of I. Value distribution of the difference operator, Arch. Math. 92 (2009) 270-278], and J. Heittokangas et al. [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J.L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009) 352-363]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.Published by Elsevier Inc.

Generalization of value distribution and uniqueness of certain types of difference polynomials

“…Lemma 2.1 (see [13]). Let f be a meromorphic function of finite order ρ and let c( = 0) be a fixed non zero complex constant.…”

Some results on zeros and uniqueness of difference-differential polynomials

“…About the uniqueness of difference products of entire functions, some results can be found in [14,15,16,18,21,27]. The main purpose is to obtain the relationships between f and g when P (f )f (z + c) and P (g)g(z + c) sharing one common value.…”

“…Combining the method of proof of [18,Lemma 5] with Lemma 2.1, we can get the following Lemma 2.4-Lemma2.7.…”

“…Recently, some papers are devoting to improve Theorem C, the constant a can be replaced by a nonzero polynomial [12] or by a small function a(z) [15]. In addition, [13,14,18,27] are devoting to the cases of meromorphic function f or more general difference products. In the following, without special stated, we assume that c is a nonzero constant, n, m, k, s, t are positive integers, a(z) is a nonzero small function with respect to f (z).…”

VALUE DISTRIBUTION AND UNIQUENESS ON q-DIFFERENCES OF MEROMORPHIC FUNCTIONS

“…Using these results, the value distribution and uniqueness of difference operators and difference polynomials of meromorphic functions have been dealt with in the past five years (see e.g. [3,4,8,10,11]). However, there are only few papers concerning with the value distribution and uniqueness of q-difference operators and q-difference polynomials (see [12,16]).…”