Geometric Properties and Applications of a Generalized Bessel–Maitland–Tremblay Function

In this paper, motivated by the modified Tremblay fractional differential operator and the normalized generalized Bessel–Maitland function, we introduce the generalized Bessel–Maitland–Tremblay function via the Hadamard product transformation. This construction established an integrated framework that connects fractional calculus with geometric function theory. By utilizing estimates for the gamma and digamma functions together with monotonicity properties of the associated coefficient sequences, we establish sufficient conditions under which the proposed function belongs to various important subclasses of normalized analytic functions. In particular, criteria are obtained for uniform convexity, starlikeness and convexity of order δ, ν-uniform starlikeness, and ν-uniform convexity, as well as exponential starlikeness, exponential convexity, and lemniscate-type conditions. Several special cases are shown to recover previously known results for the normalized generalized Bessel–Maitland function and the identity mapping. Graphical analysis and numerical examples are presented to show the geometric behavior of the proposed function and to verify the theoretical findings.