Abstract
Let k, K ∈ ℕ and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f (k) − 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most \(\nu = \left[ {\tfrac{K} {{k + 1}}} \right] \) , where ν is equal to the largest integer not exceeding \(\tfrac{K} {{k + 1}} \). In particular, if K = k, then F is normal. The results are sharp.
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References
Yang, L.: Value Distribution Theory, Springer-Verlag, Berlin, 1993
Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964
Schiff, J.: Normal Families, Springer-Verlag, Berlin, 1993
Bergweiler, W.: Bloch’s principle., Comput. Methods Funct. Theory, 6, 77–108 (2006)
Zalcman, L.: Normal families: new perspectives., Bull. Amer. Math. Soc. (N. S.), 35, 215–230 (1998)
Hayman, W. K.: Picard values of meromorphic functions and their derivatives., Ann. of Math. (2), 70, 9–42 (1959)
Hayman, W. K.: Research Problems in Function Theory, Athlone Press, London, 1967
Gu, Y. X.: A criterion for normality of families of meromorphic functions (in Chinese)., Sci. Sinica, Special Issue 1 on Math., 267–274 (1979)
Wang, Y. F., Fang, M. L.: Picard values and normal families of meromorphic functions with multiple zeros., Acta Math. Sinica (N. S.), 4, 17–26 (1998)
Nevo, S., Pang, X. C., Zalcman, L.: Quasinormality and meromorphic functions with multiple zeros., J. Anal. Math., 101, 1–23 (2007)
Xue, G. F., Pang, X. P.: A criterion for normality of a family of meromorphic functions (in Chinese)., J. East China Norm. Univ., Nat. Sci. Ed., 2, 15–22 (1988)
Pang, X. C., Zalcman, L.: Normal families and shared values., Bull. London Math. Soc., 32, 325–331 (2000)
Zalcman, L.: A heuristic principle in complex function theory., Amer. Math. Monthly, 82, 813–817 (1975)
Pang, X. C.: Bloch’s principle and normal criterion., Sci. China Ser. A, 32, 782–791 (1989)
Pang, X. C.: On normal criterion of meromorphic functions., Sci. China Ser. A, 33, 521–527 (1990)
Mirsky, L.: An Introduction to Linear Algebra, Clarendon Press, Oxford, 1955
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Supported by National Natural Science Foundation of China (Grant No. 10871094), NSFU of Jiangsu, China (Grant No. 08KJB110001), Qinglan Project of Jiangsu, China, and SRF for ROCS, SEM
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Chang, J.M. Normality and quasinormality of zero-free meromorphic functions. Acta. Math. Sin.-English Ser. 28, 707–716 (2012). https://doi.org/10.1007/s10114-011-0297-z
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DOI: https://doi.org/10.1007/s10114-011-0297-z