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A note on normal families, concerning non-exceptional functions and differential polynomials
https://doi.org/10.26907/0021-3446-2025-3-3-16
Abstract
The normality of a family of meromorphic functions whose differential polynomials satisfy a certain condition is studied. This significantly improves and generalizes some recent results of Chen (Filomat, 31 (14), 4665–4671, 2017). Moreover, the result is sharp, which is demonstrated with the help of examples.
About the Authors
N. Bharti
University of Jammu
India
India
Nikhil Bharti
Jammu, 180006
A. Singh
Maulana Azad Memorial College
India
India
Anil Singh
Jammu, 180006
References
1. Schiff J.L. Normal Families (Springer, Berlin, 1993).
2. Zalcman L. Normal families: new perspectives, Bull. Amer. Math. Soc. 35, 215–230 (1998).
3. Grahl J. Hayman’s alternative and normal families of non-vanishing meromorphic functions, Comput. Methods Funct. Theory 2 (2), 481–508 (2004).
4. Dethloff G., Tan T.V., Thin N.V. Normal criteria for families of meromorphic functions, J. Math. Anal. Appl. 411 (2), 675–683 (2014).
5. Charak K.S., Sharma S. Some normality criteria and a counterexample to the converse of Bloch’s principle, Bull. Aust. Math. Soc. 95 (2), 238–249 (2017).
6. Thin N.V. Normal family of meromorphic functions sharing holomorphic functions and the converse of the Bloch principle, Acta Math. Sci. 37 (3), 623–656 (2017).
7. Thin N.V. Normal criteria for family meromorphic functions sharing holomorphic function, Bull. Malays. Math. Sci. Soc. 40, 1413–1442 (2017).
8. Thin N.V., Oanh B.T.K. Normality criteria for families of zero-free meromorphic functions, Analysis 36 (3), 211–222 (2016).
9. Deng B.M., Fang M.L., Liu D. Normal families of zero-free meromorphic functions, J. Aust. Math. Soc. 91 (3), 313–322 (2011).
10. Chang J.M. Normality and quasinormality of zero-free meromorphic functions, Acta Math. Sin. Engl. Ser. 28 (4), 707–716 (2012).
11. Gu Y.X. A normal criterion of meromorphic families, Scientia Sinica 1, 267–274 (1979).
12. Hayman W.K. Research Problems in Function Theory (Athlone Press, London, 1967).
13. Chen J.F. Nonexceptional functions and normal families of zero-free meromorphic functions, Filomat 31 (14), 4665–4671 (2017).
14. Bharti N. A normal criterion concerning sequence of functions and their differential polynomials, Probl. Anal. Issues Anal. 13 (31) (2), 3–24 (2024).
15. Hayman W.K. Meromorphic Functions (Clarendon Press, Oxford, 1964).
16. Chen H.H., Gu Y.X. Improvement of Marty’s criterion and its application, Sci. China Ser. A 36 (6), 674–681 (1993).
17. Pang X.C., Zalcman L. Normal families and shared values, Bull. Lond. Math. Soc. 32 (3), 325–331 (2000).
18. Lahiri I., Charak K.S., Singh A. An extension of Hayman’s alternative to a class of differential polynomials, Boll. Unione. Mat. Ital. 16, 747–761 (2023), URL : https://doi.org/10.1007/s40574-023-00365-9.
Review
For citations:
Bharti N., Singh A. A note on normal families, concerning non-exceptional functions and differential polynomials. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;(3):3-16. (In Russ.) https://doi.org/10.26907/0021-3446-2025-3-3-16
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