A Picard type theorem and Bloch law
The author proves the following result for a transcendental entire function \(f\): If \(a\neq 0\) is a finite complex number and \(n\geq 2\) 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 \({\mathcal F}_ 0= \{f_ m(z)= mz: | z|< 1\}\) has the property \(f_ m+ af'{}^ n_ m\neq 0\) 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 \({\mathcal F}_ 0\) is not normal in \(D\). The following result recoups normality: If \(\mathcal F\) is a family of holomorphic functions in a domain \(D\) such that \(f\neq b\) and \(f+ af'{}^ n\neq b\) \((n\geq 2)\) for all \(f\in {\mathcal F}\), then \(\mathcal F\) is a normal family. (The interested reader is referred to a more rigorous form of the above heuristic principle due to \textit{L. Zalcman} [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)]. The family \({\mathcal F}_ 0\) above does not provide an exception to Zalcman's formulation).
scientific article; zbMATH DE number 589870