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Zbl 1158.30020
Fang, Mingliang; Zalcman, Lawrence
On the value distribution of $f +a(f\prime)^n$. (English)
[J] Sci. China, Ser. A 51, No. 7, 1196-1202 (2008). ISSN 1006-9283; ISSN 1862-2763/e

Let $f$ be a transcendental meromorphic function, and let $a$ be a nonzero finite complex number. {\it Y. Ye} [Chin. Ann. Math., Ser. B 15, No. 1, 75--80 (1994; Zbl 0801.30027)] proved that $f+a(f')^n$ assumes every finite complex value infinitely often for each positive integer $n\ge 3$. It is a question whether the result remains valid for $n=2$. The authors give an affirmative answer for this question. The authors also consider a normality criterion corresponding to the result above. Let ${\cal F}$ be a family of meromorphic functions on the plane $D$, let $n\ge 2$ be a positive integer, and let $a\ne 0$, $b$ be complex numbers. If, for each $f\in{\cal F}$, all zeros of $f$ are multiple and $f+a(f')^n\ne b$ on $D$, then ${\cal F}$ is normal on $D$. The tools for the proofs are the standard Nevanlinna theory and the rescaliug lemma due to the second author of this paper.
[Katsuya Ishizaki (Saitama)]

MSC 2000:: *30D35 Distribution of values (one complex variable)
30D45 Normal functions, etc.

Keywords: meromorphic function; normal families; differential polynomial; Nevanlinna theory

Citations: Zbl 0801.30027

Cited in: Zbl 1177.30032

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