The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces ${\mathcal{H}}^p$ and ${\mathcal{H}}^{\infty }$. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k-uniform convexity and k-uniform starlikeness. Furthermore, some sufficient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples.

The primary aim of this paper is to construct harmonic mapping associated with four parameter Wright functions. Further sufficient conditions have been established so that this harmonic mapping satisfies geometric properties such as harmonic starlikeness and convexity. Moreover, we examine relation between several subclasses of family of harmonic close-to-convexity mappings. The results obtained in this paper are novel and their significance is illustrated by several consequences.