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Normal families and uniqueness of entire functions and their derivatives. (English) Zbl 1104.30019

Let \(F\) be a family of holomorphic functions in a domain \(D\) and \(a\) and \(b\) be two complex numbers such that \(b\neq a\), \(0\). If for each \(f\) in \(F\) and \(z\) in \(D\), \(f(z)= a\) implies \(f'(z)= a\) and \(f'(z)= b\) implies \(f(z)= b\), then \(F\) is a normal family in \(D\). This non-trivial interesting result is used to prove for \(f\) a non-constant entire function, \(a\) and \(b\) two complex numbers with \(b\neq a\), \(0\), if \(f\) and \(f'\) share the value a counting multiplicity and if \(f= b\) whenever \(f= b\), then \(f\) is identically \(f'\). The latter theorem improves a result of L. Rubel and C. C. Yang [Complex Anal. Proc. Conf., Lexington 1976, Lect. Notes. Math. 599, 101–103 (1977; Zbl 0362.30026)].

MSC:

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

Citations:

Zbl 0362.30026
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