Sharing Set and Normal Families of Entire Functions

Abstract

Let \({\mathcal{F}}\) be a family of holomorphic functions defined in a domain \({\mathcal{D}}\) , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each \({f \in \mathcal{F}}\) , all zeros of f(z) are of multiplicity at least k, and f and f(k) share the set S in \({\mathcal{D}}\) , then \({\mathcal{F}}\) is normal in \({\mathcal{D}}\). As an application, we prove a uniqueness theorem.