The Normality of Meromorphic Functions with Multiple Zeros and Poles Concerning Sharing Values
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In this paper we study the problem of normal families of meromorphic func- tions concerning shared values. Let F be a family of meromorphic functions in the plane domain D ⊆ C and n, k be two positive integers such that n ≥ k + 1, and let a, b be two finite complex constants such that a ≠ 0. Suppose that (1) f + a(f(k))n and g + a(g(k))n share b in D for every pair of functions f, g ∈ F; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each f in f ∈ F D; (3) Zeros of f(k)(z) are not the b points of f(z) for each f in F ∈ D. Then F is normal in D. And some examples are provided to show the result is sharp.
Abstract
In this paper we study the problem of normal families of meromorphic func- tions concerning shared values. Let F be a family of meromorphic functions in the plane domain D ⊆ C and n, k be two positive integers such that n ≥ k + 1, and let a, b be two finite complex constants such that a ≠ 0. Suppose that (1) f + a(f(k))n and g + a(g(k))n share b in D for every pair of functions f, g ∈ F; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each f in f ∈ F D; (3) Zeros of f(k)(z) are not the b points of f(z) for each f in F ∈ D. Then F is normal in D. And some examples are provided to show the result is sharp.
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