On classes of analytic functions with respect to symmetric points defined using the Mittag-Leffler function of Le Roy type

In this article, we will define an operator using the Hadamard product of an analytic function involving the Mittag-Leffler function of the Le Roy type. The defined operator extends and unifies the well-known operators that were defined to study various subclasses of univalent functions. Using the defined operator, we introduce a new family of analytic functions with respect to symmetric points. The analytic characterization of the defined function class was mainly motivated by the so-called $\alpha$-exponentially convex functions. Further, the defined function class involves $(r,\ell )$-symmetrical function which is a generalization of the notions such as even, odd, and $\ell$-symmetrical functions. Estimates of the initial bounds, Fekete-Szegő inequality, and the integral representation of the defined function class are the main results of the paper. Applications of the main results are presented as inclusion relations and corollaries.