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Some normality criteria of function families concerning shared values. (English) Zbl 1273.30023
Summary: We study the normality of families of meromorphic functions related to shared values. We mainly consider whether a family of meromorphic functions $$\mathcal F$$ is normal in a domain $$D$$, if (i) for every pair of functions $$f$$ and $$g$$ in $$\mathcal F$$, $$f^{(k)} - af^{n}$$ and $$g^{(k)} - ag^{n}$$ share the value $$b$$, and (ii) $$f$$ has no zero of multiplicity less than $$k$$ in $$D$$ for every function $$f \in \mathcal F$$, where $$a$$ and $$b$$ are two finite complex numbers such that $$a \neq 0$$, $$n\geq k+3$$ and $$k \geq 2$$ are two positive integers. An example shows that the condition (ii) in our results is best possible.

##### MSC:
 30D45 Normal functions of one complex variable, normal families 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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##### References:
  Yang, Uniqueness Theory of Meromorphic Functions (2003) · Zbl 1070.30011  Gu, Theory of Normal Family and its Applications (2007)  Hayman, Meromorphic Functions (1964)  Yang, Value Distribution Theory (1993) · Zbl 0790.30018  Bergweiler, Bloch’s principle, Computational Methods and Function Theory 6 pp 77– (2006) · Zbl 1101.30034  Rubel, Four counterexamples to Bloch’s principle, Proceeding of the American Mathematical Society 98 pp 257– (1986) · Zbl 0602.30040  Drasin, Normal families and the Nevanlinna theory, Acta Mathematica 122 pp 231– (1969) · Zbl 0176.02802  Schwick, Normality criteria for families of meromorphic function, Journal D Analyse of Mathematique 52 pp 241– (1989) · Zbl 0667.30028  Pang, Normal families and shared values, Bulletin of the London Mathematical Society 32 pp 325– (2000) · Zbl 1030.30031  Pang, Normality and shared values, Arkiv för Matematik 38 pp 171– (2000)  Zhang, Normal families of meromorphic functions concerning shared values, Journal of Mathematical Analysis and Applications 338 pp 545– (2008) · Zbl 1145.30013  Hayman, Research Problems of Function Theory (1967)  Pang, On normal criterion of meromorphic functions, Science in China (Series A) 33 pp 521– (1990)  Chen, On a theorem of Drasin, Advances in Mathematics (China) 20 pp 504– (1991)  Ye, A new criterion and its application, Chinese Annals of Mathematics, Series A (Supplement) 2 pp 44– (1991) · Zbl 0766.30029  Chen, An improvement of Marty’s criterion and its applications, Chinese Annals of Mathematics, Series A 36 pp 674– (1993) · Zbl 0777.30018  Li, On normality criterion of a class of the functions, Journal of Fujian Normal University 2 pp 156– (1984)  Li, Proof of Hayman’s conjecture on normal families, Science in China (Series A) 28 pp 596– (1985)  Langley, On normal families and a result of Drasin, Proceedings of the Royal Society of Edinburgh A98 pp 385– (1984)  Zalcman L On some questions of Hayman 1994 5  Fang, On the normality for families of meromorphic functions, Indian Journal of Mathematics 43 (3) pp 341– (2001) · Zbl 1040.30015  Chen, The value distribution of fnf ’ , Science in China (Series A) 38 pp 121– (1995)  Xu, Normal families of meromorphic functions, Journal of Mathematics 21 (4) pp 381– (2001) · Zbl 0994.30020  Pang, Normality conditions for differential polynomials, Kexue Tongbao (in Chinese) 33 (22) pp 1690– (1988) · Zbl 1382.30060  Xu, Normality criteria of families of meromorphic functions, Indian Journal of Pure and Applied Mathematics 31 (1) pp 61– (2000) · Zbl 0943.30020  Zalcman, A heuristic principle in complex function theory, American Mathematical Monthly 82 pp 813– (1975) · Zbl 0315.30036  Zalcman, Normal families: new perspectives, Bulletin of the American Mathematical Society 35 pp 215– (1998) · Zbl 1037.30021
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