Korean J. Math. Vol. 34 No. 2 (2026) pp.165-173
DOI: https://doi.org/10.11568/kjm.2026.34.2.165

Normality criterion leading to counterexamples for the converse of Bloch's principle

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Published: Jun 30, 2026
Keywords:
Normal families, Meromorphic function, Differential inequalities, Differential Polynomial, Bloch's Principle
Mathematics Subject Classification:
30D45

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Virender Singh
Department of Mathematics, University of Jammu, Jammu-180 006, INDIA
Banarsi Lal
Department of Mathematics, School of Sciences, Cluster University of Jammu, Jammu-180 001, INDIA
Priyanka
Department of Mathematics, School of Sciences, Cluster University of Jammu, Jammu-180 001, INDIA

Abstract




In this paper we establish a normality criterion related to differential in- equalities for a family of meromorphic functions which leads to some counterexamples to the converse of Bloch’s Principle. Further, we prove a normality criteria concerning non homogeneous differential polynomial that gives a partial answer to a question posed by N. Bharti and R. Kumar[Normal families concerning partially shared functions and differential polynomials, Asian-European J. of Math., 17 12, (2024)].






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References

[1] N. Bharti and R. Kumar, Normal families concerning partially shared functions and differential polynomials, Asian-European J. Math. 17 (12) (2024), 2450082. https://doi.org/10.1142/S1793557124500827 Google Scholar

[2] K. S. Charak and J. Rieppo, Two normality criteria and the converse of the Bloch principle, J. Math. Anal. Appl. 353 (1) (2009), 43–48. https://doi.org/10.1016/j.jmaa.2008.11.066 Google Scholar

[3] K. S. Charak and V. Singh, Two normality criteria and counterexamples to the converse of Bloch’s principle, J. Kodai Math. 38 (3) (2015), 672–686. https://doi.org/10.2996/kmj/1446210601 Google Scholar

[4] J. F. Chen, Shared sets and normal families of meromorphic functions, Rocky Mountain J. Math. 40 (2) (2010), 1473–1479. [suspicious link removed] Google Scholar

[5] J. Grahl and S. Nevo, Exceptional functions wandering on the sphere and normal families, Israel J. Math. 202 (1) (2014), 21–34. https://doi.org/10.1007/s11856-014-1054-7 Google Scholar

[6] J. Grahl and S. Nevo, Spherical derivatives and normal families, J. Anal. Math. 117 (1) (2012), 119–128. https://doi.org/10.1007/s11854-012-0016-4 Google Scholar

[7] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford (1964). Google Scholar

[8] I. Lahiri, A simple normality criterion leading to a counterexample to the converse of the Bloch principle, New Zealand J. Math. 34 (2005), 61–65. Google Scholar

[9] T. Manket and S. Nevo, On row differential inequalities related to normality and quasi-normality, Comput. Methods Funct. Theory 24 (2024), 479–511. https://doi.org/10.1007/s40315-024-00524-9 Google Scholar

[10] X. C. Pang, Bloch principle and normal criterion, Sci. China Ser. A 32 (7) (1989), 782–791. Google Scholar

[11] A. Robinson, Mathematical problems, J. Symbolic Logic 38 (3) (1973), 500–516. https://doi.org/10.2307/2273049 Google Scholar

[12] L. A. Rubel, Four counterexamples to Bloch’s principle, Proc. Amer. Math. Soc. 98 (2) (1986), 257–260. https://doi.org/10.1090/S0002-9939-1986-0854029-2 Google Scholar

[13] J. L. Schiff, Normal Families, Springer-Verlag, New York (1993). https://doi.org/10.1007/978-1-4612-0907-2 Google Scholar

[14] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813–817. https://doi.org/10.1080/00029890.1975.11993942 Google Scholar