Geometric analysis of integral operators related to Touchard polynomials and generalized Bessel functions

This article investigated the inclusion properties of Touchard polynomials $Y_m(p,z)$ and generalized Bessel functions of the first kind ${\mathcal{G}}_a(z)$, along with their associated convolution and integral operators, within the analytic function classes ${\mathcal{R}}_{s,d}^w(\delta )$ and ${\mathcal{M}}_{b,s}^w(\delta )$. Using coefficient bounds of certain function classes, we derived sufficient conditions for the convolution operators ${\mathcal{Y}}_m^p(\psi ,z)$ and ${\mathcal{E}}_{l_a,c}(\psi ,z)$, and the integral operators ${\mathfrak{Y}}_m(p,z)$ and ${\mathfrak{G}}_a(z)$ to belong to various subclasses of starlike and convex functions. Furthermore, inclusion results among these subclasses were obtained, thereby extending and unifying several existing results in geometric function theory.