Uniqueness Problems on Meromorphic Functions which Share Four Small Functions

Abstract

Throughout this note the term “meromorphic” means meromorphic in the whole complex plane, and we shall use the standard notation of the Nevanlinna theory of meromorphic functions, see e.g. [3], [7]. In particular, S(r, f) denote any quantity that satisfies S(r, f) = o(1)T(r,f) as r → ∞, possibly outside a set of r of finite linear measure for a transcendental meromorphic function f. A meromorphic function a is called “small” with respect to f if T(r, a) = S(r, f) holds. We say that two transcendental meromorphic functions f and g share a small function a provided that f(z) - a(z)= 0 if and only if g(z) - a(z) = 0 in the case a(z) ≢ ∞. We also consider the case a(z)≡ ∞. If every pole of f is a pole of g and vice versa, we call f and g share ∞. For the sake of simplicity, we call z 0 is an a-point of f if f (z 0) - a(z 0) = 0, and the multiplicity of a-point is derived from the multiplicity of zero of f (z) - a(z). We state that f and g share a small function a CM (counting multiplicities), if a is shared by f and g and if a k-fold a-point z 0 of f is also a k-fold a-point of g, k = k(z 0). In contract to sharing CM we also use the notation sharing IM (ignoring multiplicities).KeywordsMeromorphic functionSharing problemSmall functio