Uniqueness of Meromorphic Functions with Their n th Order Differences
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Uniqueness of Meromorphic Functions with Their nth Order Differences Xiaoguang Qi1
· Lianzhong Yang2
Received: 6 May 2020 / Revised: 23 July 2020 / Accepted: 19 August 2020 © Iranian Mathematical Society 2020
Abstract Let f be a transcendental meromorphic function of hyper-order strictly less than 1. In this paper, we deal with the uniqueness problem on f sharing two values with its nth order differences n f . And this research extends earlier results by Chen and Yi (Result Math 63:557–565, 2013), Gao et al. (Anal Math 45:321–334, 2019) and Lü and Lü (Comput Methods Funct Theory 17:395–403, 2017). Keywords Meromorphic functions · Value sharing · Difference operator Mathematics Subject Classification 30D35 · 39A10
1 Introduction and Main Results In this paper, we shall use the basic concepts of Nevanlinna Theory [19]. In particular, we denote difference as c f (z) = f (z + c) − f (z) and nc f (z) = n noperators n−1 n−i f (z + ic), where n(≥ 2) is an integer and c c (c f (z)) = i=0 i (−1)
Communicated by Ali Abkar. The work was supported by the NNSF of China (nos. 11661052, 11801215) and the NSF of Shandong Province (nos. ZR2016AQ20, ZR2018MA021).
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Xiaoguang Qi [email protected]; [email protected] Lianzhong Yang [email protected]
1
School of Mathematics, University of Jinan, Jinan 250022, Shandong, People’s Republic of China
2
School of Mathematics, Shandong University, Jinan 250100, Shandong, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
is a non-zero constant. Moreover, if c = 1, then we use the difference notations c f (z) = f (z) and nc f (z) = n f (z). The study of uniqueness problems should originate from the following two classical results due to Nevanlinna [16] Theorem 1.1 If two meromorphic functions f (z) and g(z) share five distinct values a1 , a2 , a3 , a4 , a5 ∈ C ∪ {∞} IM, then f (z) = g(z). Theorem 1.2 If two meromorphic functions f (z) and g(z) share four distinct values a1 , a2 , a3 , a4 ∈ C ∪ {∞} CM, then f (z) = g(z) or f (z) = T ◦ g, where T is a Möbius transformation. It is well known that “4 CM”can not be replaced by “4 IM”, see [6]. Furthermore, Gundersen [7, Theorem 1] weakened the assumption “4 CM”by “2 CM + 2 IM”, however, “1 CM + 3 IM” still remains open. If g(z) is the derivative of f (z), then we can get some further results on functions sharing values with their derivatives. For the related results, the reader is invited to see [19, Chapter 8]. Recently, Heittokangas et al. [11,12] obtained some uniqueness results when g(z) is the shift of f (z). In fact, the study on this uniqueness problem is carried out with the development of the difference version to the usual Nevanlinna theory, especially the difference-type logarithmic derivative lemma, which starts in the papers [4,8,9]. Theorem 1.3 Let f (z) be a meromorphic function of finite order, let c ∈ C, and let a1 , a2 , a3 ∈ S( f ) ∪ {∞} be three distinct periodic functions with period c. If f (z) and f (z + c) share a1 , a2 CM and a3 IM, then f (z) = f (z + c). Remark 1.
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