Meromorphic functions that weighted sharing three values and one pair

Abstract

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.

On results of G. Brosch and T.C. Alzahary

“…Let A 1 ≡ 0. If A 3 ≡ 0, then from (3.23) we get β = A 2 A 4 , which contradicts (3.20) and if A 4 ≡ 0, then from (3.23) we get g = A 2 A 3 , which implies T (r, g) = S(r), a contradiction. So A 3 ≡ 0 and…”

“…If f and g assume (xii) of Theorem 1.1, then f − a = − (2e γ +1) 2 3(1+e γ +e 2γ ) 2 and g − b = (2e γ +1)(e γ −1) 3(1+e γ +e 2γ ) . If f and g assume (xiii) of Theorem 1.1, then f − a = (e γ +2) 2 3(1+e γ +e 2γ ) and g − b = (e γ +2)(e γ −1) 3(1+e γ +e 2γ ) . If f and g assume (xiv) of Theorem 1.…”

“…11) where A = 3C − B2 4 − B . Clearly T (r, A) + T (r, B) + T (r, C) = S(r) and since w = F F and F = (f − a) g−f f (g−1) , we get by Lemma 2.1 and (3.8) that S(r, w) = S(r).Let H ≡ 0.…”

The uniqueness theorems of meromorphic functions sharing three values and a set with two elements

“…where γ is a nonconstant entire function, s and k ( 2) are positive integers such that s and k + 1 are mutually prime and 1 s k. [12].) Let f be a nonconstant meromorphic function, and let …”

On a result of G. Brosch

“…Let f , g share (0,1),(1, m),(∞, k) and f ≡ g, where (m − 1)(mk − 1) > (1 + m) 2 . If α = f −1 g−1 and β = g f then N(r, a; α) + N(r, a; β) = S(r) for a = 0, ∞.…”