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On the value distribution of $f+a{\left(f\text{'}\right)}^{n}$. (English) Zbl 1158.30020
Let $f$ be a transcendental meromorphic function, and let $a$ be a nonzero finite complex number. Y. Ye [Chin. Ann. Math., Ser. B 15, No. 1, 75–80 (1994; Zbl 0801.30027)] proved that $f+a{\left({f}^{\text{'}}\right)}^{n}$ 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 $ℱ$ 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 $f\in ℱ$, all zeros of $f$ are multiple and $f+a{\left({f}^{\text{'}}\right)}^{n}\ne b$ on $D$, then $ℱ$ 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.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 30D45 Bloch functions, normal functions, normal families
##### References:
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