In this paper, we introduce a new linear operator involving the $q$-analogue of generalized Bessel functions and the $q$-analogue of the Carlson-Shaffer operator. By using this operator, we define and investigate certain subclasses of analytic functions in the open unit disk. We obtain coefficient estimates and establish Fekete-Szegö-type inequalities for the newly introduced function classes. Moreover, by employing the Hadamard product, we study several geometric properties and derive necessary as well as sufficient conditions for these classes. The results obtained in this work generalize and extend a number of earlier results in the existing literature.

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.