Geometric Properties of Certain Entire Functions Associated with the Rogers - Szegó and Stieltjes - Wigert Polynomials

In this paper, we investigate the geometric properties of two families of entire functions which are closely related to the Rogers–Szegő and Stieltjes–Wigert polynomials. By exploiting the Laguerre–Pólya class structure and Hadamard factorization, we determine the radii of starlikeness and convexity of order β ϵ [0,1), as well as the radii of uniform convexity for several natural normalizations. Since these radii are characterized as the smallest positive zeros of certain transcendental equations, we derive explicit lower and upper bounds via Euler–Rayleigh inequalities expressed in terms of the Maclaurin coefficients. Numerical experiments and graphical representations are included to illustrate the sharpness of the obtained bounds and the associated geometric behavior. The results contribute to the geometric function theory of $q$-series by providing a unified analytic framework for these classes of entire functions. MSC 2020: 30C45, 33D15, 30C15.