*(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 ${\mathcal{F}}_{0}=\{{f}_{m}\left(z\right)=mz:\left|z\right|<1\}$ has the property ${f}_{m}+a{f}^{\text{'}}{}_{m}^{n}\ne 0$ in $D:\left|z\right|<1$ 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 $f\ne b$ and $f+a{f}^{\text{'}}{}^{n}\ne b$ $(n\ge 2)$ 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 *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).

##### MSC:

30D30 | General theory of meromorphic functions |

30D45 | Bloch functions, normal functions, normal families |