On Certain Class of Meromorphic Functions with Positive Coefficients Associated with Mittag-Leffler Function Based on Hilbert Space Operator

The Mittag-Leffler~\cite{mit} function ascends naturally in the solution of fractional order differential and integral equations, and exclusively in the studies of fractional generalizing of kinetic equation, random walks, L\'{e}vy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class of meromorphic functions defined in the punctured unit disk $\Delta^*:= \{z\in\mathbb{C}: 0<|z|<1\}$ based on Mittag-Leffler function denoted by $\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$. We discuss its characteristic properties like coefficient inequalities, growth and distortion inequalities, as well as closure results for $f\in\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$extensively. 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 prove a Property using an integral operator and its inverse defined on the new class.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 its characteristic properties by specializing the parameters which are new and not studied in association with Mittag-Leffler functions.