Small functions and weighted sharing three values

Abstract

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.

Small functions and uniqueness of meromorphic functions

“…Let f and g be two distinct non-constant meromorphic functions sharing 0, 1, ∞ CM. In the paper we give a complete answer to the question of Alzahary and also improve Theorem C. [2] exhibited that Theorem D does not hold for transcendental small functions. We show that this small function is in fact only a special type of transcendental small function for which Theorem D does not hold.…”

“…Theorem C. (See [2].) Let f and g be two distinct non-constant meromorphic functions sharing (0, 1), (1, ∞), (∞, ∞) and let a = a(z) ( ≡ 0, 1, ∞) be a small meromorphic function of f and g. If N(r, a; g) = T (r, g) + S(r, g), then N(r, a; g) = S(r, g) and there exists a non-constant meromorphic function γ , such that f and g satisfy one of the following three possibilities:…”

“…Theorem D. (See [2].) Let f and g be two distinct non-constant meromorphic functions sharing 0, 1, ∞ CM.…”

“…(See [2].) Let g 1 and g 2 be non-constant meromorphic functions such that g 1 g 2 is not a constant.…”

“…In [2] Alzahary asked the following question: Let f and g be two distinct non-constant meromorphic functions sharing 0, 1, ∞ CM. Also suppose that a = a(z) ( ≡ 0, 1, ∞) is a non-constant small function of f and g. For which kind of function a = a(z) do we have N(r, a; f ) = T (r, f ) + S(r, f ) and N(r, a; g) = T (r, g) + S(r, g)?…”