Uniqueness of meromorphic functions sharing a meromorphic function of a smaller order with their derivatives. (English) Zbl 1201.30034

Li, Xiao-Min; Yi, Hong-Xun

Ann. Pol. Math. 98, No. 3, 207-219 (2010).

Five uniqueness theorems for meromorphic functions and their derivatives are obtained. One of the results is the following: Let \(f(z)\) be a meromorphic function of finite order, and let \(a(z)\) be a meromorphic function satisfying \(\sigma(a)<\sigma(f)\). Assume that \(f(z)\) and \(a(z)\) have finitely many poles and have no common poles. If \(f(z)- a(z)\) and \(f^{(k)}(z)- a(z)\) share \(0\) CM, where \(k\geq 1\) is a positive integer, then \(f^{(k)}(z)- a(z)= c\big(f(z)- a(z)\big)\) for some constant \(c\). This result is a generalization of [J. M. Chang and Y. Z. Zhu, J. Math. Anal. Appl. 351, No. 1, 491–496 (2009; Zbl 1162.30021)]. The tools of the proof are from Nevanlinna theory and Wiman-Valiron theory.

Reviewer: Katsuya Ishizaki (Saitama)

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30	Meromorphic functions of one complex variable (general theory)

Keywords:

meromorphic functions; sharing values; sharing small functions; Wiman-Valiron theory

Citations:

Zbl 1162.30021