The author proves the following result for a transcendental entire function : If is a finite complex number and is an integer, then 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 is apt to be a normal family in if the property cannot be possessed by a non-constant entire function. The family has the property in 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 is not normal in . The following result recoups normality: If is a family of holomorphic functions in a domain such that and for all , 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 above does not provide an exception to Zalcman’s formulation).