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On the value distribution of f+a(f') 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(f ' ) n assumes every finite complex value infinitely often for each positive integer n3. 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 be a family of meromorphic functions on the plane D, let n2 be a positive integer, and let a0, b be complex numbers. If, for each f, all zeros of f are multiple and f+a(f ' ) n b on D, then 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.

30D35Distribution of values (one complex variable); Nevanlinna theory
30D45Bloch functions, normal functions, normal families
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