On uniqueness of \(L\)-functions in terms of zeros of strong uniqueness polynomial. (English) Zbl 1529.11097
Summary: In this article, we have mainly focused on the uniqueness problem of an \(L\)-function \(\mathcal{L}\) with an \(L\)-function or a meromorphic function \(f\) under the condition of sharing the sets, generated from the zero set of some strong uniqueness polynomials. We have introduced two new definitions, which extend two existing important definitions of URSM and UPM in the literature and the same have been used to prove one of our main results. As an application of the result, we have exhibited a much improved and extended version of a recent result of
H. H. Khoai et al. [“On value distribution of L-functions sharing finite sets with meromorphic functions”, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 66(114), No. 3, 265–280 (2023)]. Our remaining results are about the uniqueness of \(L\)-function under weighted sharing of sets generated from the zeros of a suitable strong uniqueness polynomial, which improve and extend some results in
[H. H. Khoai and V. H. An, Ramanujan J. 58, No. 1, 253–267 (2022; Zbl 1492.30078)].
MSC:
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
Keywords:
meromorphic function; strong uniqueness polynomial; uniqueness; shared sets; \(L\) functionCitations:
Zbl 1492.30078References:
| [1] | T. T. H. An, J. T.-Y. Wang, and P.-M. Wong, “Strong uniqueness polynomials: the complex case,” Complex Var. Theory Appl., vol. 49, no. 1, pp. 25-54, 2004, doi: 10.1080/02781070310001634601. · Zbl 1077.30024 |
| [2] | A. Banerjee and S. Maity, “Further investigations on a unique range set under weight 0 and 1,” Carpathian Math. Publ., vol. 14, no. 2, pp. 504-512, 2022. · Zbl 1508.30061 |
| [3] | A. Banerjee, “Uniqueness of meromorphic functions sharing two sets with finite weight II,” Tamkang J. Math., vol. 41, no. 4, pp. 379-392, 2010. · Zbl 1213.30052 |
| [4] | A. Banerjee and I. Lahiri, “A uniqueness polynomial generating a unique range set and vice versa,” Comput. Methods Funct. Theory, vol. 12, no. 2, pp. 527-539, 2012, doi: 10.1007/BF03321842. · Zbl 1283.30064 |
| [5] | A. Banerjee and S. Mallick, “On the characterisations of a new class of strong uniqueness polynomials generating unique range sets,” Comput. Methods Funct. Theory, vol. 17, no. 1, pp. 19-45, 2017, doi: 10.1007/s40315-016-0174-y. · Zbl 1364.30035 |
| [6] | H. Fujimoto, “On uniqueness of meromorphic functions sharing finite sets,” Amer. J. Math., vol. 122, no. 6, pp. 1175-1203, 2000. · Zbl 0983.30013 |
| [7] | H. Fujimoto, “On uniqueness polynomials for meromorphic functions,” Nagoya Math. J., vol. 170, pp. 33-46, 2003, doi: 10.1017/S0027763000008527. · Zbl 1046.30011 |
| [8] | F. Gross, “Factorization of meromorphic functions and some open problems,” in Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), ser. Lecture Notes in Math. Springer, Berlin-New York, 1977, vol. 599, pp. 51-67. · Zbl 0357.30007 |
| [9] | W. K. Hayman, Meromorphic functions, ser. Oxford Mathematical Monographs. Clarendon Press, Oxford, 1964. · Zbl 0115.06203 |
| [10] | P.-C. Hu and B. Q. Li, “A simple proof and strengthening of a uniqueness theorem for L- functions,” Canad. Math. Bull., vol. 59, no. 1, pp. 119-122, 2016, doi: 10.4153/CMB-2015- 045-1. · Zbl 1334.30001 |
| [11] | H. H. Khoai, V. H. An, and L. Q. Ninh, “Value-sharing and uniqueness for L-functions,” Ann. Polon. Math., vol. 126, no. 3, pp. 265-278, 2021, doi: 10.4064/ap201030-17-3. · Zbl 1482.30087 |
| [12] | H. H. Khoai and V. H. An, “Determining an L-function in the extended Selberg class by its preimages of subsets,” Ramanujan J., vol. 58, no. 1, pp. 253-267, 2022, doi: 10.1007/s11139- 021-00483-y. · Zbl 1492.30078 |
| [13] | H. H. Khoai, V. H. An, and N. D. Phuong, “On value distribution of L-functions sharing finite sets with meromorphic functions,” Bull. Math. Soc. Sci. Math. Roumanie (N.S.), vol. 66(114), no. 3, pp. 265-280, 2023. |
| [14] | I. Lahiri, “Weighted value sharing and uniqueness of meromorphic functions,” Complex Variables Theory Appl., vol. 46, no. 3, pp. 241-253, 2001, doi: 10.1080/17476930108815411. · Zbl 1025.30027 |
| [15] | P. Li and C.-C. Yang, “Some further results on the unique range sets of meromorphic functions,” Kodai Math. J., vol. 18, no. 3, pp. 437-450, 1995, doi: 10.2996/kmj/1138043482. · Zbl 0849.30025 |
| [16] | P. Lin and W. Lin, “Value distribution of L-functions concerning sharing sets,” Filomat, vol. 30, no. 14, pp. 3795-3806, 2016, doi: 10.2298/FIL1614795L. · Zbl 1474.30222 |
| [17] | A. Z. Mohon’ko, “The Nevanlinna characteristics of certain meromorphic functions,” Teor. Funkcii Funkcional. Anal. i Priložen., no. 14, pp. 83-87, 1971. · Zbl 0237.30030 |
| [18] | A. Selberg, “Old and new conjectures and results about a class of Dirichlet series,” in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989). Univ. Salerno, Salerno, 1992, pp. 367-385. · Zbl 0787.11037 |
| [19] | J. Steuding, Value-distribution of L-functions, ser. Lecture Notes in Mathematics. Springer, Berlin, 2007, vol. 1877. · Zbl 1130.11044 |
| [20] | C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, ser. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 2003, vol. 557, doi: 10.1007/978-94-017-3626-8. · Zbl 1070.30011 |
| [21] | H.-X. Yi, “The reduced unique range sets for entire or meromorphic functions,” Complex Variables Theory Appl., vol. 32, no. 3, pp. 191-198, 1997, doi: 10.1080/17476939708814990. · Zbl 0881.30026 |
| [22] | Q.-Q. Yuan, X.-M. Li, and H.-X. Yi, “Value distribution of L-functions and uniqueness questions of F. Gross,” Lith. Math. J., vol. 58, no. 2, pp. 249-262, 2018, doi: 10.1007/s10986-018- 9390-7. · Zbl 1439.11231 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
