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A Picard type theorem and Bloch law. (English) Zbl 0801.30027

The author proves the following result for a transcendental entire function f: If a0 is a finite complex number and n2 is an integer, then f+af ' n assumes all finite complex numbers infinitely often.

A well-known heuristic function theoretic principle asserts that a family of holomorphic functions which have a common property in a domain D is apt to be a normal family in D if the property cannot be possessed by a non-constant entire function. The family 0 ={f m (z)=mz:|z|<1} has the property f m +af ' m n 0 in D:|z|<1 which, by the above result, cannot hold for a non- constant entire function. The author notes that this yields an exception to the above principle since the family 0 is not normal in D. The following result recoups normality: If is a family of holomorphic functions in a domain D such that fb and f+af ' n b (n2) for all f, then is a normal family.

(The interested reader is referred to a more rigorous form of the above heuristic principle due to L. Zalcman [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)]. The family 0 above does not provide an exception to Zalcman’s formulation).


MSC:
30D30General theory of meromorphic functions
30D45Bloch functions, normal functions, normal families
Keywords:
normal family