q ‐Poisson and q ‐Pascal Distribution Series for Analytic Function Classes

In this paper, we study analytic function classes defined by the q ‐derivative and convolution operators generated by discrete q ‐distributions. We first consider coefficient conditions for the class determined by a q ‐derivative inequality and its negative‐coefficient subclass. We then investigate the q ‐Poisson and q ‐Pascal distribution series and derive explicit sufficient conditions for their membership in this class. In addition, we establish inclusion results for convolutions with q ‐starlike and q ‐convex functions and prove corresponding self‐inclusion properties. Numerical examples are provided to verify the theoretical results. These results connect distribution‐based convolution operators with q ‐derivative inequalities in geometric function theory.