TAILIEUCHUNG - Difference uniqueness theorems on meromorphic functions in several variable
In this paper, we mainly investigate the uniqueness problem on meromorphic functions in Cm sharing small functions with their difference operators or shifts, and we obtain some interesting results that act as some extensions of previous results from one complex variable to several complex variables. | Turk J Math (2018) 42: 2481 – 2505 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Difference uniqueness theorems on meromorphic functions in several variables Zhixue LIU,, Qingcai ZHANG∗, Department of Mathematics, School of Information, Renmin University of China, Beijing, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we mainly investigate the uniqueness problem on meromorphic functions in Cm sharing small functions with their difference operators or shifts, and we obtain some interesting results that act as some extensions of previous results from one complex variable to several complex variables. Key words: Uniqueness, meromorphic functions, difference operators, several variables 1. Introduction and main results Let f be a meromorphic function in the complex domain. In this paper, we assume that the reader is familiar with standard notations such as characteristic function T (r, f ) , counting function N (r, f ), and fundamental results of the Nevanlinna theory of meromorphic functions (see [8, 12, 17]). We say that α(z) is a small function with respect to f if T (r, α) = o(T (r, f )) as r → ∞ outside of a possible exceptional set of finite logarithmic measure. Usually, S(f ) is used to denote the family of all small functions with respect to f . For two meromorphic functions f, g , if f − α and g − α have the same zeros, counting multiplicity (ignoring multiplicity), then f and g share the small function α CM (IM). In the last decades, uniqueness problems on meromorphic functions have been studied deeply due to their important value in Nevanlinna theory, and many interesting results have been established (see [9, 13, 16, 18]). As a very active subject, the problems on uniqueness of the entire function sharing values with its derivatives were initiated by Rubel and Yang[15]. In 1986, Jank et al.[10] .
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