The linear dependency of L−functions and meromorphic functions sharing finite sets
Main Article Content
Abstract
In this paper we investigate the linear dependency of L functions and meromorphic functions sharing finite sets. As a consequence, we present some classes of subsets S,T in C such that for a meromorphic function f and an L-function L, the condition that f and L share S and T, respectively (counting multiplicity) implies f = hL for a non-zero constant h. We discuss some applications of main result. The main result obtained in this paper improves and extends a recent result due to the authors in [32]. We extend previous results of Yuan, Li and Yi [32] by considering distinct finite sets S,T and establishing linear dependency between f and L. Our results are inspired by a work of Yuan, Li, and Yi in [32] and Khoai et al. in [11] and [13].
Article Details
Keywords
$L$-function, linear dependency, shared sets, Meromorphic function
References
[1] V. H. An and P. Chanthaphone, Uniqueness of L-functions sharing finite sets with meromorphic functions having Deficient Poles, J. Math. Math. Sci., Vol. 3, No. 2, pp. 1-15, 2024.
[2] V. H. An and P. Chanthaphone, Value distribution of L-functions and meromorphic functions sharing finite sets, South East Asian J. of Mathematics and Mathematical Sciences, Vol. 20, No. 3, pp. 147-164, 2024.
[3] A. Banerjee and A. Kundu, On uniqueness of L-functions in terms of zeros of strong uniqueness polynomial, Cubo, A Mathematical Journal, Vol. 25, no. 03, pp. 497-514, 2023.
[4] T. Dinh, Ensemble d’unicit´e pour les polynˆomes, Ergodic Theory Dynam. Systems, 22:1, 171-186, 2022.
[5] A. A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, vol. 236, 2008.
[6] F. Gross, Factorization of meromorphic functions and some open problems, Complex Analysis (Proc. Conf . Univ. Kentucky, Lexington, Ky.1976), pp. 51-69, Lecture Notes in Math. Vol. 599, Springer, Berlin, 1977.
[7] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122, 1175-1203, 2000.
[8] W. K. Hayman, Meromorphic Functions, Clarendon, Oxford (1964).
[9] A. Kundu and A. Banerjee, Sufficient conditions to determine the linear dependency of two meromorphic functions, Mathematica bohemica, Doi: 10.21136/Mb.2024.0140-23.
[10] H. H. Khoai, V. H. An and N. X. Lai, Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions, Int. J. Math., Vol. 29, N. 5, pp. 122-140, 2018.
[11] H. H. Khoai, V. H. An, andL. Q. Ninh, Value-sharing and uniqueness for L-functions, Ann. Polon. Math., vol. 126, no. 3, pp. 265-278, 2021.
[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.
[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, vol. 66(114), no. 3, pp. 265-280, 2023.
[14] J. Kaczorowski, G. Molteni, A. Perelli, Linear independence of L functions, Forum Math., 18, 1-7, 2006.
[15] X.-M. Li and H.-X. Yi, Meromorphic functions sharing three values, J. Math. Soc. Japan, Vol. 56, No. 1, 147-167, 2024.
[16] P.-C. Hu and B. Q. Li, A simple proof and strengthening of a uniqueness theorem for L-functions, Canad. Math. Bull., 59, 119-122, 2016.
[17] B.Q. Li, A result on value distribution of L-functions, Proc. Amer. Math. Soc., Vol. 138, N. 6, 2071-2077, 2010.
[18] B.Q. Li, A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation, Adv. Math., 226, 4198-4211, 2011.
[19] P. Lin and W. Lin, Value distribution of L-functions concerning sharing sets, Filomat, 30:16, 3975-3806, 2016.
[20] Y. Li, W. Lin, Set sharing results for derivatives of meromorphic functions, J. Math. Res. Appl. 42, 587–598, 2022.
[21] N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Springer, 2017.
[22] J. Steuding, Value-Distribution of L-functions, Lecture Notes in Mathematics, 1877, Springer, 2007.
[23] P. Sahoo and S. Halder, Results on L-functions and certain uniqueness questions of Gross, Lith. Math. J., 60(6), 80-91, 2020.
[24] I. Ostrovskii, F. Pakovitch and M. Zaidenberg, A remark on complex polynomials of least derivation, Internat. Math. Res. Notices, 14, 699-703,1996.
[25] F. Pakovich, On polynomials sharing preimages of compact sets, and related questions, Geometric and Functional Analysis 18(1), 163-183, 2008.
[26] C. C. Yang, Open problems, in: S. S. Miller (ed.),Complex Analysis (Brockport,NY, 1976), Lecture Notes in Pure Appl. Math. 36, Dekker, New York, 169-170, 1978.
[27] C. C. Yang, On deficiencies of differential polynomials, Math. Z. 116, 197-204 (1970).
[28] C. C. Yang, On deficiencies of differential polynomials, Math. Z. 125, 107-112 (1972).
[29] C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Kluwer Acad. Publ. (2003).
[30] A.-D. Wu and P.-C. Hu, Uniqueness theorems for Dirichlet series, Bull. Aust. Math. Soc. 91, 389-399, 2015.
[31] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192, 225–294, 2024.
[32] Q.-Q. Yuan, X.-M. Li, and H.-X. Yi, Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2), 249-262, 2018.
[2] V. H. An and P. Chanthaphone, Value distribution of L-functions and meromorphic functions sharing finite sets, South East Asian J. of Mathematics and Mathematical Sciences, Vol. 20, No. 3, pp. 147-164, 2024.
[3] A. Banerjee and A. Kundu, On uniqueness of L-functions in terms of zeros of strong uniqueness polynomial, Cubo, A Mathematical Journal, Vol. 25, no. 03, pp. 497-514, 2023.
[4] T. Dinh, Ensemble d’unicit´e pour les polynˆomes, Ergodic Theory Dynam. Systems, 22:1, 171-186, 2022.
[5] A. A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, vol. 236, 2008.
[6] F. Gross, Factorization of meromorphic functions and some open problems, Complex Analysis (Proc. Conf . Univ. Kentucky, Lexington, Ky.1976), pp. 51-69, Lecture Notes in Math. Vol. 599, Springer, Berlin, 1977.
[7] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122, 1175-1203, 2000.
[8] W. K. Hayman, Meromorphic Functions, Clarendon, Oxford (1964).
[9] A. Kundu and A. Banerjee, Sufficient conditions to determine the linear dependency of two meromorphic functions, Mathematica bohemica, Doi: 10.21136/Mb.2024.0140-23.
[10] H. H. Khoai, V. H. An and N. X. Lai, Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions, Int. J. Math., Vol. 29, N. 5, pp. 122-140, 2018.
[11] H. H. Khoai, V. H. An, andL. Q. Ninh, Value-sharing and uniqueness for L-functions, Ann. Polon. Math., vol. 126, no. 3, pp. 265-278, 2021.
[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.
[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, vol. 66(114), no. 3, pp. 265-280, 2023.
[14] J. Kaczorowski, G. Molteni, A. Perelli, Linear independence of L functions, Forum Math., 18, 1-7, 2006.
[15] X.-M. Li and H.-X. Yi, Meromorphic functions sharing three values, J. Math. Soc. Japan, Vol. 56, No. 1, 147-167, 2024.
[16] P.-C. Hu and B. Q. Li, A simple proof and strengthening of a uniqueness theorem for L-functions, Canad. Math. Bull., 59, 119-122, 2016.
[17] B.Q. Li, A result on value distribution of L-functions, Proc. Amer. Math. Soc., Vol. 138, N. 6, 2071-2077, 2010.
[18] B.Q. Li, A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation, Adv. Math., 226, 4198-4211, 2011.
[19] P. Lin and W. Lin, Value distribution of L-functions concerning sharing sets, Filomat, 30:16, 3975-3806, 2016.
[20] Y. Li, W. Lin, Set sharing results for derivatives of meromorphic functions, J. Math. Res. Appl. 42, 587–598, 2022.
[21] N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Springer, 2017.
[22] J. Steuding, Value-Distribution of L-functions, Lecture Notes in Mathematics, 1877, Springer, 2007.
[23] P. Sahoo and S. Halder, Results on L-functions and certain uniqueness questions of Gross, Lith. Math. J., 60(6), 80-91, 2020.
[24] I. Ostrovskii, F. Pakovitch and M. Zaidenberg, A remark on complex polynomials of least derivation, Internat. Math. Res. Notices, 14, 699-703,1996.
[25] F. Pakovich, On polynomials sharing preimages of compact sets, and related questions, Geometric and Functional Analysis 18(1), 163-183, 2008.
[26] C. C. Yang, Open problems, in: S. S. Miller (ed.),Complex Analysis (Brockport,NY, 1976), Lecture Notes in Pure Appl. Math. 36, Dekker, New York, 169-170, 1978.
[27] C. C. Yang, On deficiencies of differential polynomials, Math. Z. 116, 197-204 (1970).
[28] C. C. Yang, On deficiencies of differential polynomials, Math. Z. 125, 107-112 (1972).
[29] C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Kluwer Acad. Publ. (2003).
[30] A.-D. Wu and P.-C. Hu, Uniqueness theorems for Dirichlet series, Bull. Aust. Math. Soc. 91, 389-399, 2015.
[31] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192, 225–294, 2024.
[32] Q.-Q. Yuan, X.-M. Li, and H.-X. Yi, Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2), 249-262, 2018.