Some unity results on entire functions and their difference operators related to 4 CM theorem
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(2020) 2020:220
RESEARCH
Open Access
Some unity results on entire functions and their difference operators related to 4 CM theorem BaoQin Chen1 and Sheng Li1,2* *
Correspondence: [email protected] Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhangjiang, 524088, P.R. China 2 Southern Marine Science and Engineering Guangdong Laboratory (Zhanjiang), Zhanjiang 524088, China 1
Abstract This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let f (z) be a nonconstant entire function of finite order, and let a1 , a2 be two distinct finite complex constants. If f (z) and nη f (z) share a1 and a2 “CM”, then f (z) ≡ nη f (z), and hence f (z) and nη f (z) share a1 and a2 CM. MSC: 30D35; 39B32 Keywords: Entire functions; Differences; Shared values; Nevanlinna theory
1 Introduction and main results It is well known that a monic polynomial is uniquely determined by its zeros and a rational function by its zeros and poles ignoring a constant factor. But it becomes much more complicated to deal with the transcendental meromorphic function case. In 1929, Nevanlinna proved his famous 5 IM theorem and 4 CM theorem (see e.g. [20, 23]): if meromorphic functions f (z) and g(z) share five (respectively, four) distinct values in the extended complex plane IM (respectively, CM), then f (z) ≡ g(z) ((respectively, f (z) = T(g(z)), where T is a Möbius transformation). Here and in what follows, we say that f (z) and g(z) share the finite value a CM(IM) if f (z) – a and g(z) – a have the same zeros with the same multiplicities (ignoring multiplicities), and we say that f (z) and g(z) share the ∞ CM(IM) if f (z) and g(z) have the same poles with the same multiplicities (ignoring multiplicities). To relax those shared conditions in Nevanlinna’s 4 CM theorem, Gundersen provided an example to show that 4 CM shared values cannot be replaced with 4 IM shared values, but with 3 CM shared values and 1 IM shared value in [5]. That is, “4 IM = 4 CM” and “3 CM + 1 IM = 4 CM”. In addition, he showed that “2 CM + 1 IM = 4 CM” in [6] (see correction in [8]), as well as by Mues in [17]. The problem that “1 CM + 3 IM = 4 CM” is still open. We recall the following result by Mues in [17], which mainly inspired us to write this paper. Theorem A ([17]) Let f and g be nonconstant meromorphic functions sharing four distinct values aj (j = 1, 2, 3, 4) “CM”. If f ≡ g, then f and g share aj (j = 1, 2, 3, 4) CM. © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless ind
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