On pseudo-Mittag-Leffler functions and applications

The Mittag-Leffer function and related distribution are valuable in statistics, applied mathematics and lifetime data analysis. In this paper, we introduce the concept of pseudo-Mittag-Leffler functions. Then, we discuss the pseudo-fractional boundary value problems in two classes of pseudo-Caputo fractional derivative and pseudo-Riemann-Liouville fractional derivative to get the pseudo-integral inequalities of Lyapunov-type. Also, as an application, we obtain the nonexistence of real zeros of pseudo-Mittag-Leffler functions. Our results generalize the corresponding ones in the literature. Finally, two open problems for further investigations are given.