## 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

Zbl 0362.30026
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