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Exceptional functions and normal families of holomorphic functions with multiple zeros
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Jun-Fan Chen
Published/Copyright:
April 6, 2011
Abstract
Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.
Received: 2008-06-10
Published Online: 2011-04-06
Published in Print: 2011-March
© de Gruyter 2011
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Keywords for this article
Normal family;
exceptional function;
holomorphic function;
meromorphic function
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