Unicity of meromorphic functions related to their derivatives

Abstract

AbstractIn this paper, we shall study the unicity of meromorphic functions definedover non-Archimedean fields of characteristic zero such that their valence func-tions of poles grow slower than their characteristic functions. If f is such afunction, and f and a linear differential polynomial P(f) of f, whose coeffi-cients are meromorphic functions growing slower than f, share one finite valuea CM, and share another finite value b (6= a) IM, then P(f) = f. 1 Introduction. In 1929, R. Nevanlinna studied the unicity of meromorphic functions in C. Thefive value theorem due to R. Nevanlinna states that if two non-constant meromor-phic functions f and gin C share five distinct complex numbers a j IM (ignoringmultiplicity), which meansf −1 (a j ) = g −1 (a j ), j= 1,2,...,5in the sense of sets, then it follows that f = g. The four value theorem of R.Nevanlinna states that if two non-constant meromorphic functions f and gin Cshare four distinct complex numbers a j CM (counting multiplicity), which meansf