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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.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
30D45Bloch functions, normal functions, normal families
References:
[1]Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
[2]Yang L. Value Distribution Theory. Berlin Heidelberg: Springer-Verlag; Beijing: Science Press, 1993
[3]Hayman W K. Picard values of meromorphic functions and their derivatives. Ann of Math, 70(2): 9–42 (1959) · Zbl 0088.28505 · doi:10.2307/1969890
[4]Mues E. Über ein Problem von Hayman. Math Z, 164: 239–259 (1979) · Zbl 0402.30034 · doi:10.1007/BF01182271
[5]Ye Y S. A Picard type theorem and Bloch law. Chinese Ann Math Ser B, 15: 75–80 (1994)
[6]Bergweiler W. Bloch’s principle. Comput Methods Funct Theory, 6: 77–108 (2006)
[7]Pang X C. On normal criterion of meromorphic functions. Sci China Ser A-Math, 33: 521–527 (1990)
[8]Zalcman L. New light on normal families. In: Zalcman L, ed. Proceedings of the Ashkelon Workshop on Complex Function Theory, May, 1996. Bar-Ilan University, 1997, 237–245
[9]Yang L. Precise fundamental inequalities and sum of deficiencies. Sci China Ser A-Math, 34: 157–165 (1991)
[10]Zalcman L. Normal families: new perspectives. Bull Amer Math Soc (NS), 35: 215–230 (1998) · Zbl 1037.30021 · doi:10.1090/S0273-0979-98-00755-1
[11]Chen H H, Gu Y X. Improvement of Marty’s criterion and its application. Sci China Ser A-Math, 36: 674–681 (1993)
[12]Pang X C, Zalcman L. Normal families and shared values. Bull London Math Soc, 32: 325–331 (2000) · Zbl 1030.30031 · doi:10.1112/S002460939900644X