The aim of this paper is to investigate certain properties such as convexity of order $\mu$ , close-to-convexity of order $\left(1+\mu \right)$ /2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag–Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy space ${\mathcal{H}}^p$ . Some of our results improve the results in the literature.