Abstract: The Mittag-Leffler function arises naturally in solving differential and integral equations of fractional order and especially in the study of fractional generalization of kinetic equation, random walks, Levy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\) of meromorphic functions defined in the punctured unit disk \(\Delta^*:= \{z\in\mathbb{C}: 0<|z|<1\}\) based on Mittag-Leffler. We discuss extensively its characteristic properties like coefficient inequalities, growth and distortion inequalities, as well as closure results for \(f\in\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\). Properties of a certain integral operator and its inverse defined on the new class \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\) are also discussed. Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. We also establish some results concerning neighborhoods and the partial sums of meromorphic functions in this new class. We also state some new subclasses and their characteristic properties by specializing the parameters which are new and not studied before in association with Mittag-Leffler functions.
Keywords: meromorphic functions, starlike function, convolution, positive coefficients, coefficient inequalities, integral operator, Mittag-Leffler function, Hilbert space operator
For citation: Murugusundaramoorthy, G. and Vijaya, K. On a New Class of Meromorphic Functions Associated with Mittag-Leffler Function, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 70-86. DOI 10.46698/p1426-1765-3037-f
1. Dziok, J., Murugusundaramoorthy, G. and Sokol, J. On Certain Class of Meromorphic Functions with Positive Coefficients, Acta Mathematica Scientia, 2012, vol. 32, no. 4, pp. 1376-1390. DOI: 10.1016/S0252-9602(12)60106-4.
2. Clunie, J. On Meromorphic Schlicht Functions, Journal of the London Mathematical Society, 1959, vol. s1-34, no. 2, pp. 215-216. DOI: 10.1112/jlms/s1-34.2.215.
3. Nehari, Z. and Netanyahu, E. On the Coefficients of Meromorphic
Schlicht Functions, Proceedings of the American Mathematical Society, 1957, vol. 8, pp. 15-23. DOI: 10.1090/S0002-9939-1957-0083038-0.
4. Pommerenke, Ch. Uber Einige Klassen Meromorpher Schlichter
Funktionen, Mathematische Zeitschrift, 1962, vol. 78, pp. 263-284. DOI: 10.1007/BF01195174.
5. Pommerenke, Ch. On Meromorphic Starlike Functions, Pacific Journal of Mathematics, 1963, vol. 13, pp. 221-235.
6. Royster, W. C. Meromorphic Starlike Multivalent Functions, Transactions of the American Mathematical Society, 1963, vol. 107, pp. 300-308. DOI: 10.2307/1993896.
7. Bajpai, S. K. A Note on a Class of Meromorphic Univalent Functions, Revue Roumaine de Matheematiques Pures et Appliquees, 1997, vol. 22, pp. 295-297.
8. Mogra, M. L., Reddy, T. R. and Juneja, O. P. Meromorphic Univalent Functions with Positive Coefficients, Bulletin of the Australian Mathematical Society, 1985, vol. 32, no. 2, pp. 161-176.
DOI: 10.1017/S0004972700009874.
9. Uralegaddi, B. A. and Ganigi, M. D. A Certain Class of Meromorphically Starlike Functions with
Positive Coefficients, Pure and Applied Mathematika Sciences, 1987, vol. 26,
pp. 75-81.
10. Cho, N. E., Lee, S. H. and Owa, S. A Class of Meromorphic Univalent Functions with
Positive Coefficients, Kobe Journal of Mathematics, 1987, vol. 4, no. 1, pp. 43-50.
11. Aouf, M. K. On a Certain Class of Meromorphic Univalent Functions
with Positive Coefficients, Rendiconti di Matematica e delle sue Applicazioni, 1991, ser. 11, no. 2, pp. 209-219.
12. Uralegaddi, B. A. and Somanatha, C. Certain Differential Operators for
Meromorphic Functions, Houston Journal of Mathematics, 1991, vol. 17, no. 2,
pp. 279-284.
13. Duren, P. L. Univalent Functions, Grundlehren der Mathematischen
Wissenschaften, vol. 259, Springer-Verlag, New York, 1983.
14. Srivastava, H. M. and Owa, S. Current Topics in Analytic Function Theory, New Jersey, World Scientific Publishing, 1992.
15. Mittag-Leffler, G. M. Sur la Nouvelle Fonction \(\mathbf{E(x)}\), Comptes Rendus de l'Academie des Sciences, Paris, 1903, vol. 137, pp. 554-558.
16. Wiman, A. Uber den Fundamentalsatz in der Teorie der Funktionen \(\mathbf{E_a(x)}\), Acta Mathematica, 1905, vol. 29, pp. 191-201. DOI: 10.1007/BF02403202.
17. Attiya, A. A. Some Applications of Mittag-Leffler Function in the Unit Disk, Filomat, 2016, vol. 30, no. 7, pp. 2075-2081. DOI: 10.2298/FIL1607075A.
18. Bansal, D. and Prajapat, J. K. Certain Geometric Properties of the Mittag-Leffler Functions, Complex Variables and Elliptic Equations, 2016, vol. 61, no. 3, pp. 338-350. DOI: 10.1080/17476933.2015.1079628.
19. Frasin, B. A. An Application of an Operator Associated with Generalized Mittag-Leffler Function, Konuralp Journal of Mathematics, 2019, vol. 7, no. 1, pp. 199-202.
20. Frasin, B. A., Tariq Al-Hawary, and Yousef, F. Some Properties of a Linear Operator Involving Generalized Mittag-Leffler Function, Studia Universitatis Babes-Bolyai. Chemia, 2020, vol. 65, no. 1, pp. 67-75. DOI: 10.24193/subbmath.2020.1.06.
21. Haubold, H. J., Mathai, A. M. and Saxena, R. K. Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, 2011. DOI: 10.1155/2011/298628.
22. Kiryakova, V. Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; co-published in the United States with John Wiley & Sons, Inc., New York, 1994.
23. Srivastava, H. M. and Tomovski, Z. Fractional Calculus with an Itegral Operator Containing Generalized Mittag-Leffler Function in the Kernal, Applied Mathematics and Computation, 2009, vol. 211, no. 1, pp. 198-210.
24. Aouf, M. K. and Mostafa, A. Certain Inequalities of Meromorphic Univalent
Functions Associated with the Mittag-Lefer Function, Journal of Applied Analysis,
2019, vol. 25, no. 2, pp. 173-178. DOI: 10.1515/jaa-2019-0018.
25. Bao, G. J. and Ling, Y. On Some Classes of Analytic Functions
with Negative Coefficients, Journal of Harbin Institute of Technology, 1991, no. 23, pp. 100-103.
26. Dunford, N. and Schwartz, J. T. Linear Operators. Part I: General Theory (Reprinted
from the 1958 original), A Wiley-Interscience Publication, John Wiley & Sons, New York, 1988.
27. Auof, M. K. and Murugusundaramoorthy, G. On a Subclass of Uniformly Convex Functions Defined by the Dziok-Srivastava Operator, Australian Journal of Mathematical Analysis and Applications, 2008, vol. 5, no. 1, article 3, pp. 1-17.
28. Liu, J.-L. and Srivastava, H. M. Classes of Meromorphically Multivalent
Functions Associated with the Generalized Hypergeometric Function, Mathematical and Computer Modelling, 2004, vol. 39, no. 1, pp. 21-34. DOI: 10.1016/S0895-7177(04)90503-1.
29. Liu, J.-L. and Srivastava, H. M. Subclasses of Meromorphically
Multivalent Functions Associated with a Certain Linear Operator, Mathematical and Computer Modelling, 2004, vol. 39, no. 1, pp. 35-44.
DOI: 10.1016/S0895-7177(04)90504-3.
30. Owa, S. and Pascu, N. N. Coefficient Inequalities for Certain
Classes of Meromorphically Starlike and Meromorphically Convex
Functions, Journal of Inequalities in Pure and Applied Mathematics, 2003, vol. 4, no. 1, article
17, pp. 1-6.
31. Murugusundaramoorthy, G. and Rosy, T. Subclass of Analytic Functions Associated
with Fox-Wright's Generalized Hypergeometric Functions Based on Hilbert Space
Operator, Studia Universitatis Babes-Bolyai Math., 2011, vol. 56, no. 3, pp. 61-72.
32. Akgul, A. A New Subclass of Meromorphic Functions Defined by Hilbert Space Operator, Honam Mathematical Journal, 2016, vol. 38, no. 3, pp. 495-506. DOI: 10.5831/HMJ.2016.38.3.495.
33. Akgul, A. and Bulut, S. On a Certain Subclass of Meromorphic Functions Defined by Hilbert Space Operator, Acta Universitatis Apulensis, 2016, no. 45, pp. 1-9. DOI: 10.17114/j.aua.2016.45.01.
34. Goodman, A. W. Univalent Functions and Nonanalytic Curve, Proceedings of the American
Mathematical Society, 1957, vol. 8, no. 3, pp. 598-601. DOI: 10.2307/2033525.
35. Ruscheweyh, S. Neighborhoods of Univalent Funtions, Proceedings of the American Mathematical Society, 1981, vol. 81, no. 4, pp. 521-527.
36. Silverman, H. Partial Sums of Starlike and Convex
Functions, Journal of Mathematical Analysis and Applications, 1997, vol. 209, no. 1, pp. 221-227.
DOI: 10.1006/jmaa.1997.5361.
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
