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Zbl 0801.30027
Ye, Yasheng
A Picard type theorem and Bloch law. (English)
[J] Chin. Ann. Math., Ser. B 15, No. 1, 75-80 (1994). ISSN 0252-9599; ISSN 1860-6261/e

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 $f+ af'{}\sp n$ assumes all finite complex numbers infinitely often.\par 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 ${\cal F}\sb 0= \{f\sb m(z)= mz: \vert z\vert< 1\}$ has the property $f\sb m+ af'{}\sp n\sb m\ne 0$ in $D: \vert z\vert< 1$ 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 ${\cal F}\sb 0$ is not normal in $D$. The following result recoups normality: If $\cal F$ is a family of holomorphic functions in a domain $D$ such that $f\ne b$ and $f+ af'{}\sp n\ne b$ $(n\ge 2)$ for all $f\in {\cal F}$, then $\cal F$ is a normal family.\par (The interested reader is referred to a more rigorous form of the above heuristic principle due to {\it L. Zalcman} [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)]. The family ${\cal F}\sb 0$ above does not provide an exception to Zalcman's formulation).
[S.Dragosh (East Lansing)]

MSC 2000:: *30D30 General theory of meromorphic functions
30D45 Normal functions, etc.

Keywords: normal family

Citations: Zbl 0315.30036

Cited in: Zbl 1158.30020

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