Using the idea of weighted sharing, we prove some results on uniqueness of meromorphic functions sharing three values which improve some results given by H.-X. Yi, I. Lahiri, X. Hua and other authors.  2004 Published by Elsevier Inc.

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G. Brosch improved the theorem of Nevanlinna for four values theorem and proved that let f and g be two nonconstant meromorphic functions sharing 0, 1, y CM, and let a and b be two finite complex numbers such that a; b B f0; 1g. If f ¼ a , g ¼ b, then f is a fractional linear transformation of g. In this paper we extend this theorem by using the idea of weighted sharing.Definition 1 (see [6, p. 189]). Let p be a positive integer, we denote by N pÞ ðr; f Þ (or N pÞ ðr; f Þ) the counting function of poles of f with multiplicities a p (ignoring multiplicities). We further define 13 2000 Mathematics Subject Classification: 30D35, 30D30.

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and 1 CM, and a is a nonconstant rational function, then N 2Þ ðr, 1=ðg À aÞÞ ¼ Tðr, gÞ þ Sðr, gÞ and N 2Þ ðr, 1=ð f À aÞÞ ¼ Tðr, f Þ þ Sðr, f Þ: An example shows that the latter result is not true for some transcendental small functions of f and g.