# Normality and uniqueness of homogeneous differential polynomials

### Abstract

The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.

### References

J. B. Conway, Functions of one complex variable, Springer-Verlag, 1973.

W. K. Hayman, Meromorphic Functions, Oxford University Press, 1964.

I. Lahiri, P. Sahoo, On a uniqueness theorem of H. Ueda, J. Korean Math. Soc., 47 (2010), no.3, 467–482.

I. Laine, Nevanlinna Theory and complex differential equations, Walter de Gruyter, Berlin, 1993.

F. Lu and H.X. Yi, The Bruck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc., 48, 499-512, 2011.

S. Majumder, A result on a conjecture of W. Lu, Q. Li and C. Yang, Bull. Korean Math. Soc., 53 (2016), 411–421.

S. Majumder, An open problem of L¨u, Li and Yang, Filomat, 33 (2019), no.1, 163–175.

E. Mues, N. Steinmetz, Meromorphic funktionen, die mit ableitung werte teilen, Manuscripta Math., 29 (1979), 195–206.

L. A. Rubel, C. C. Yang, Values shared by an entire function and its derivatives, Lecture Notes in Mathematics, 599, Springer-Verlag, Berlin, 1977, 101–103.

W. Bergweiler, A new proof of the Ahlfors five islands theorem, J. Anal. Math., 76 (1998), 337–347. https://doi.org/10.1007/BF02786941

P. Sahoo, G. Biswas, Uniqueness of entire function sharing polynomials with their derivatives, Kyungpook Math. J., 58 (2018), 519–531.

N. Shilpa, L. N. Achala, Uniqueness of meromorphic functions of a certain non-linear differential polynomials, Int. Elec. J. Pure Appl. Math., 10 (2016), no.1, 23–39.

J.F. Xu, H.X. Yi, A precise inequality of differential polynomials related to small functions, J. Math. Ineq., 10 (2016), no.4, 971–976.

C.C. Yang, H.X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic publishers, 2003.

L.Z. Yang, J.L. Zhang, Non-existence of meromorphic solutions of fermat type functional equation, Aequations Math., 76 (2008), 140–150.

H. X. Yi, On characteristic function of a meromorphic function and its derivative, Indian J. Math., 33 (1991), 119–133.

Q.C. Zhang, Meromorphic functions sharing three values, Indian J. Pure Appl. Math., 30 (1999), 667–682.

Y. Jiang, B. Huang, A note on the value distribution of fl(f(k))n, Hiroshima Math. J., 46 (2016), no.2, 135–147.

B. Chakraborty, W. Lu, On the value distribution of a differential monomials and some normality criteria, Mat. Stud., 56 (2021), no.1, 55–60.

M. Tejuswini, N. Shilpa, R. S. Dyavanal and B. Narasimha Rao, Conjecture of Lu, Li and Yang concerning differential monomials, Electron. J. Math. Anal. Appl., 10 (2022), 137–143.

N. Li, L.Z. Yang, Meromorphic function that shares one small functions with its differential polynomial, Kyungpook Math. J., 50 (2010), 447–454.

S.S. Bhoosnurmath, B. Chakraborty, H. M. Srivastava, A note on the value distribution of differential polynomials, Commun. Korean Math. Soc., 34 (2019), no.4, 1145–1155.

J. Schiff, Normal families, Springer, New York, 1993.

*Matematychni Studii*,

*59*(2), 168-177. https://doi.org/10.30970/ms.59.2.168-177

Copyright (c) 2023 R. S. Dyavanal, S. B. Kalakoti

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Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.