Uniqueness results for differential polynomials sharing a set
Soniya Sultana, Pulak Sahoo
Received July 28, 2023. Published online June 17, 2024.
Abstract: We investigate the uniqueness results of meromorphic functions if differential polynomials of the form $(Q(f))^{(k)}$ and $(Q(g))^{(k)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q'$. The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).
Keywords: uniqueness; differential polynomials; set sharing; small function
References: [1] T. T. H. An, N. V. Phuong: Uniqueness theorems for differential polynomials sharing a small function. Comput. Methods Funct. Theory 17 (2017), 613-634. DOI 10.1007/s40315-017-0198-y | MR 3712522 | Zbl 1382.30059
[2] T. T. H. An, N. V. Phuong: A lemma about meromorphic functions sharing a small function. Comput. Methods Funct. Theory 22 (2022), 277-286. DOI 10.1007/s40315-021-00388-3 | MR 4432482 | Zbl 1493.30061
[3] V. H. An, H. H. Khoai: On uniqueness for meromorphic functions and their $n$-th derivatives. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 47 (2018), 117-126. MR 3849194 | Zbl 1413.30122
[4] W. Bergweiler, A. Eremenko: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11 (1995), 355-373. DOI 10.4171/RMI/176 | MR 1344897 | Zbl 0830.30016
[5] S. S. Bhoosnurmath, R. S. Dyavanal: Uniqueness and value-sharing of meromorphic functions. Comput. Math. Appl. 53 (2007), 1191-1205. DOI 10.1016/j.camwa.2006.08.045 | MR 2327673 | Zbl 1170.30011
[6] H. Chen, M. Fang: The value distribution of $f^nf'$. Sci. China, Ser. A 38 (1995), 789-798. MR 1360682 | Zbl 0839.30026
[7] W. K. Hayman: Picard values of meromorphic functions and their derivatives. Ann. Math. (2) 70 (1959), 9-42. DOI 10.2307/1969890 | MR 0110807 | Zbl 0088.28505
[8] W. K. Hayman: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[9] H. H. Khoai, V. H. An: Uniqueness problem for meromorphic functions when two differential polynomials share a set of roots of unity. Adv. Stud.: Euro-Tbil. Math. J. 15 (2022), 39-51. DOI 10.32513/asetmj/19322008203 | MR 4425975 | Zbl 1492.30075
[10] I. Laine: Nevanlinna Theory and Complex Differential Equations. de Gruyter Studies in Mathematics 15. Walter de Gruyter, Berlin (1993). DOI 10.1515/9783110863147 | MR 1207139 | Zbl 0784.30002
[11] E. Mues: Über ein Problem von Hayman. Math. Z. 164 (1979), 239-259. (In German.) DOI 10.1007/BF01182271 | MR 0516609 | Zbl 0402.30034
[12] N. V. Phuong: Normality and uniqueness property of meromorphic function in terms of some differential polynomials. Vietnam J. Math. 49 (2021), 1317-1332. DOI 10.1007/s10013-020-00460-w | MR 4319553 | Zbl 1477.30028
[13] M. Ru: Nevanlinna Theory and Its Relation to Diophantine Approximation. World Scientific, Singapore (2001). DOI 10.1142/4508 | MR 1850002 | Zbl 0998.30030
[14] C.-C. Yang: Question 1.8. Problems in complex function theory. Complex Analysis: Proceeding of the S.U.N.Y. Brockport Conference. Lecture Notes in Pure and Applied Mathematics 36. Marcel Dekker, New York (1978), 169-170.
[15] C.-C. Yang, X. Hua: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn., Math. 22 (1997), 395-406. MR 1469799 | Zbl 0890.30019
[16] C.-C. Yang, H.-X. Yi: Uniqueness Theory of Meromorphic Functions. Mathematics and Its Applications 557. Kluwer Academic, Dordrecht (2003). MR 2105668 | Zbl 1070.30011
[17] L. Zalcman: Normal families: New perspectives. Bull. Am. Math. Soc., New Ser. (1998), 35 215-230. DOI 10.1090/S0273-0979-98-00755-1 | MR 1624862 | Zbl 1037.30021
[18] J.-L. Zhang, L.-Z. Yang: Some results related to a conjecture of R. Brück. JIPAM, J. Inequal. Pure Appl. Math. 8 (2007), Article ID 18, 11 pages. MR 2295712 | Zbl 1136.30009
Affiliations: Soniya Sultana, Department of Mathematics, Berhampore Girls' College, Shankar Mandal Rd, Gora Bazar, Berhampore, West Bengal 742101, India, e-mail: soniyasultana3@gmail.com; Pulak Sahoo (corresponding author), Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India, e-mail: sahoopulak1@gmail.com
