Meromorphic Functions Sharing a Small Function

Abstract

We will study meromorphic functions that share a small function, and prove the following result: let f(z) and g(z) be two transcendental meromorphic functions in the complex plane and let n≥11 be a positive integer. Assume that a(z)(≢0) is a common small function with respect to f(z) and g(z). If fnf′ and gng′ share a(z) CM, then either fn(z)f′(z)gn(z)g′(z)≡a2(z), or f(z)≡tg(z) for a constant satisfying tn+1=1. As applications, we give several examples.