Normal Families and Shared Values of Meromorphic Functions

Abstract

Abstract. Some criteria for determining the normality of the family F of meromorphicfunctions in the unit disc, which share values depending on f ∈ F with their derivativesis obtained. The new results in this paper improve some earlier related results given byPang and Zalcman [3], Fang and Zalcman [2], A. P. Singh and A. Singh [5]. 1. Introduction, definitions and main resultsLet f and g be meromorphic functions on a domain D in C, and let a and b becomplex numbers. If g(z) = b whenever f(z) = a, we writef(z) = a ⇒ g(z) = b.In a different notation, we have E f (a) ⊂ E g (b), where E h (c) = h −1 (c) ∩ D = {z ∈D : h(z) = c}. If f(z) = a ⇒ g(z) = b and g(z) = b ⇒ f(z) = a, we writef(z) = a ⇔ g(z) = b.If f(z) = a ⇔ g(z) = a, we say that f and g share a on D.Schwick is probably the first to find a connection between the normality crite-rion and shared values of meromorphic functions. He proved the following theorem.Theorem A([4]). Let F be a family of meromorphic functions in the unit disc ∆,and let a