New results on applications of Nevanlinna methods to value sharing problems

Abstract

Let $\mathbb{K}$ be a complete algebraically closed p-adic field of characteristic zero. We give a new Nevanlinna-type theorem that lets us obtain results of uniqueness for two meromrphic functions inside a disk, sharing 4 bounded functions CM. Let P be a polynomial of uniqueness for meromorphic functions in $\mathbb{K}$ or in an open disk, let f, g be two transcendental meromorphic functions in the whole field $\mathbb{K}$ or meromorphic functions in an open disk of $\mathbb{K}$ that are not quotients of bounded analytic functions and let α be a small meromorphic function with respect to f and g. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove a result of uniqueness for functions: we show that if f′P′(f) and g′P′(g) share α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by a new Hypothesis (G). Another consists of using the new Nevanlinna-type Theorem.