Uniqueness of Meromorphic Functions and Question of Gross

Abstract

Abstract. In this paper, we deal with the uniqueness of meromorphic functions con-cerning one question of Gross (see [5, Question 6]), and obtain some results that areimprovements of that of former authors. Moreover, the example shows that the result issharp. 1. Introduction and main resultsIn this paper, the term \meromorphic will always mean meromorphic in thecomplex plane C. We assume that the reader is familiar with the basic results andnotations of Nevanlinna's value distribution theory (see [6]), such as T (r;f ), N (r;f )and m (r;f ). Meanwhile, we need the following notations. Let f (z) be a meromor-phic function. We denote by n 1) (r;f ) the number of simple poles of f in jzj · r,N 1) (r;f ) is de¯ned in terms of n 1) (r;f ) in the usual way (see [19]). We furtherde¯ne± 1) (1 ; f ) = 1 i limsup r !1 N 1) (r;f )T (r;f ):By the de¯nition of N 1) (r;f ), we haveN 1) (r;f ) · N (r;f ) ·12N 1) (r;f )+12N (r;f ) ·12N 1) (r;f )+12T (r;f ):From this we obtain(1)12± 1) (1 ;f ) ·12±