Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T10:19:01.906Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME NORMALITY CRITERIA AND A COUNTEREXAMPLE TO THE CONVERSE OF BLOCH’S PRINCIPLE

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  23 November 2016

KULDEEP SINGH CHARAK  and
SHITTAL SHARMA
KULDEEP SINGH CHARAK*
Affiliation:
Department of Mathematics, University of Jammu, Jammu-180 006, India email [email protected]
SHITTAL SHARMA
Affiliation:
Department of Mathematics, University of Jammu, Jammu-180 006, India email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we prove some value distribution results which lead to normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family, and improve some recent results. In particular, the main result of this paper leads to a counterexample to the converse of Bloch’s principle.

Keywords

value distribution theorynormal familiesmeromorphic functionsdifferential polynomialssharing

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 30D30: Meromorphic functions, general theory 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 238 - 249
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The work of the second author is supported by University Grants Commission (UGC), India (No. F.17-77/08(SA-1)).

References

Bergweiler, W., ‘Bloch’s principle’, Comput. Methods Funct. Theory 6 (2006), 77–108.Google Scholar
Charak, K. S. and Rieppo, J., ‘Two normality criteria and converse of the Bloch principle’, J. Math. Anal. Appl. 353 (2009), 43–48.Google Scholar
Charak, K. S. and Sharma, S., ‘Value distribution and normality of meromorphic functions – involving partial sharing of small functions’, Turkish J. Math., to appear.Google Scholar
Charak, K. S. and Sharma, S., ‘A value distribution result leading to normality criteria’, Southeast Asian Bull. Math., to appear.Google Scholar
Charak, K. S. and Singh, V., ‘Two normality criteria and counterexamples to the converse of Bloch’s principle’, Kodai Math. J. 38 (2015), 672–686.Google Scholar
Dethloff, G., Tan, T. V. and Thin, N. V., ‘Normal criteria for families of meromorphic functions’, J. Math. Anal. Appl. 411 (2014), 675–683.Google Scholar
Drasin, D., ‘Normal families and Nevanlinna theory’, Acta Math. 122 (1969), 231–263.CrossRefGoogle Scholar
Dutt, G. and Kumar, S., ‘Some normality criteria’, Turkish J. Math., to appear.Google Scholar
Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
Jiang, Y. and Gao, Z., ‘Normal families of meromorphic functions sharing a holomorphic function and the converse of the Bloch principle’, Acta Math. Sci. 32B (2012), 1503–1512.Google Scholar
Lahiri, I., ‘A simple normality criterion leading to a counter example to the converse of the Bloch principle’, New Zealand J. Math. 34 (2005), 61–65.Google Scholar
Li, B. Q., ‘On the Bloch heuristic principle, normality and totally ramified values’, Arch. Math. 87 (2006), 417–426.Google Scholar
Montel, P., ‘Sur les suites infinies de fonctions’, Ann. Sci. Éc. Norm. Supér. 24 (1907), 233–334.Google Scholar
Rubel, L. A., ‘Four counterexamples to Bloch’s principle’, Proc. Amer. Math. Soc. 98 (1986), 257–260.Google Scholar
Schiff, J. L., Normal Families (Springer, Berlin, 1993).Google Scholar
Schwick, W., ‘Sharing values and normality’, Arch. Math. 59 (1992), 50–54.Google Scholar
Singh, A. P., ‘On order of homogeneous differential polynomials’, Indian J. Pure Appl. Math. 16 (1985), 791–795.Google Scholar
Xu, J. and Cao, W., ‘Some normality criteria of meromorphic functions’, J. Inequal. Appl. 2010 (2010), 926302, 10 pages.Google Scholar
Zalcman, L., ‘Normal families: new perspectives’, Bull. Amer. Math. Soc. 35 (1998), 215–230.Google Scholar