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
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