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Yosida functions and Picard values of integral functions and their derivatives. (English) Zbl 0879.30018
The following result is proven. Theorem 1: If \(f(z)\) is a transcendental integral (meromorphic) function with only zeros of order at least \(k+1\), then \(f^{(k)}(z)\) assumes every finite non-zero complex value infinitely often. This result bears on work of J. Clunie concerning a conjecture of W. Hayman for transcendental meromorphic functions. A corresponding criterion for normality is also given. Theorem 2: If, for every function \(f(z)\) in a family of holomorphic functions, \(f(z)\) has only zeros of order at least \(k+1\) and \(f^{(k)}\) does not assume the value 1, then the family is normal. The proofs of these results rely on variations of now familiar characterizations of normal families. These variations employ the use of non-constant Yosida functions as limits rather than the usual non-constant meromorphic functions. Finally, the value distribution of \(f^{2} + af^{(k)}\) is studied when \(f\) is a transcendental integral function.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
Full Text: DOI
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