Abstract
Schwick (J Anal Math 52:241–289, 1989) states that let \(\mathcal {F}\) be a family of meromorphic functions on a domain D and if for each \(f\in \mathcal {F}\), \((f^n)^{(k)}\ne 1\), for \(z\in D\), where n, k are positive integers such that \(n\ge k+3\), then \(\mathcal {F}\) is a normal family in D. In this paper we investigate the opposite view that if for each \(f\in \mathcal {F}\), \((f^n)^{(k)}(z)-\psi (z)\) has zeros in D, where \(\psi (z)\) is a holomorphic function in D, then what can be said about the normality of the family \(\mathcal {F}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill, New York (1979)
Dethloff, G., Tan, T.V., Thin, N.V.: Normal criteria for families of meromorphic functions. J. Math. Anal. Appl. 411, 675–683 (2014)
Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)
Pang, X.C., Yang, D., Zalcman, L.: Normal families and omitted functions. Indiana Univ. Math. J. 54, 223–235 (2005)
Schiff, J.: Normal Families. Springer-Verlag, Berlin (1993)
Schwick, W.: Normality criteria for families of meromophic functions. J. Anal. Math 52, 241–289 (1989)
Xu, Y.: Normal families and exceptional functions. J. Math. Anal. Appl. 329, 1343–1354 (2007)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Science Press/Kluwer Academic Publishers, Dordrecht (2003)
Yang, L.: Value Distribution Theory. Springer-Verlag, Berlin (1993)
Zalcman, L.: A heuristic principle in complex function theory. Am. Math. Monthly 82, 813–817 (1975)
Zalcman, L.: Normal families: new perspectives. Bull. Am. Math. Soc. 35(3), 215–230 (1998)
Zeng, S., Lahiri, I.: A normality criterion for meromorphic functions. Kodai Math. J. 35, 105–114 (2012)
Zeng, S., Lahiri, I.: A normality criterion for meromorphic functions having multiple zeros. Ann. Polon. Math. 110, 3 (2014)
Zhang, G., Pang, X.C., Zalcman, L.: Normal families and omitted functions II. Bull. London Math. Soc. 41, 63–71 (2009)
Zhao, L., Wu, X.: Normal families of holomorphic functions and multiple values. Bull. Belg. Math. Soc. Simon Stevin 19, 535–547 (2012)
Acknowledgments
We wish to thank Indrajit Lahiri (Kalyani University) and Kaushal Verma (IISc Bangalore) for their valuable suggestions and help.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
The research work of the first author is supported by research fellowship from UGC India.
Rights and permissions
About this article
Cite this article
Datt, G., Kumar, S. A normality criterion for a family of meromorphic functions. Monatsh Math 180, 193–204 (2016). https://doi.org/10.1007/s00605-016-0896-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0896-y