Geometric Properties, Partial Sums, and Fekete–Szegö Inequalities for a Normalized Four-Parameter Le Roy-Type q-Mittag–Leffler Function

In this paper, we introduce and study a normalized four-parameter Le Roy-type q-Mittag–Leffler (LR-q-ML) function defined by means of the q-Pochhammer symbol, the q-Gamma function, and the q-factorial, whose parameters directly influence the associated coefficient sequence and its growth behavior. We establish monotonicity properties of the related coefficient sequences, which are then used to derive coefficient estimates, Fekete–Szegö inequalities, upper bounds for the normalized function and its q-difference operator, and lower bounds for the real parts of the ratios involving the normalized function and its partial sums. Numerical computations together with graphical illustrations verify the theoretical estimates and show that the partial sums provide accurate approximation to the complete function as the number of terms increases. The obtained results provide a unified framework for investigating geometric properties and coefficient results of higher-parameter q-special functions in geometric function theory.