Close-to-convexity and partial sums for normalized Le Roy-type $ q $-Mittag-Leffler functions

In recent years, researchers have explored the properties of close-to-convexity and partial sums for various Mittag-Leffler functions, including $ q $-Mittag-Leffler, Bernas Mittag-Leffler, and Le Roy-type Mittag-Leffler functions. Building on previous research, this paper explores the Le Roy-type $ q $-Mittag-Leffler function in the open unit disk, with a specific focus on its normalization. We also use the concept of $ q $-close-to-convex functions, and examine whether the Le Roy-type $ q $-Mittag-Leffler function possesses close-to-convexity. Moreover, we determine lower bounds for the normalized Le Roy-type $ q $-Mittag-Leffler and its sequence of partial sums. We also present some lemmas, propositions, examples, and meaningful corollaries that highlight the importance of our findings. The results presented in this paper are new and demonstrate improvements over some existing findings in the literature.