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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 $a\ne 0$ is a finite complex number and $n\ge 2$ is an integer, then $f+a{f}^{\text{'}}{}^{n}$ 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 ${ℱ}_{0}=\left\{{f}_{m}\left(z\right)=mz:|z|<1\right\}$ has the property ${f}_{m}+a{f}^{\text{'}}{}_{m}^{n}\ne 0$ 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 ${ℱ}_{0}$ is not normal in $D$. The following result recoups normality: If $ℱ$ is a family of holomorphic functions in a domain $D$ such that $f\ne b$ and $f+a{f}^{\text{'}}{}^{n}\ne b$ $\left(n\ge 2\right)$ for all $f\in ℱ$, then $ℱ$ 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 ${ℱ}_{0}$ above does not provide an exception to Zalcman’s formulation).

##### MSC:
 30D30 General theory of meromorphic functions 30D45 Bloch functions, normal functions, normal families
normal family