Comprehensive Inequalities and Equations Specified by the Mittag-Leffler Functions and Fractional Calculus in the Complex Plane

Abstract

Inequalities or equations appertaining to (generalized) Mittag-Leffler functions and/or asserted by (generalized) fractional calculus play important roles in themselves and also in their diverse applications in nearly all sciences and engineering. Many inequalities or equations involving (one variable and three parameters of) the Mittag-Leffler (type) functions and also (generalized) fractional calculus have been established by several researchers in many different ways. In this investigation, many comprehensive results containing several differential inequalities and/or equations (in the complex plane \(\mathbb {C}\)) in relation with (one variable and three parameters of) the Mittag-Leffler (type) functions given by $$E_{\alpha ,\beta }^{\gamma }(z):=\sum _{n=0}^{\infty } \frac{(\gamma )_n}{n!\ \! \Gamma (n\alpha +\beta )}z^n \ \ \ \big ( \beta ,\gamma \in \mathbb {C};\mathfrak {R}e(\alpha )>0\big ),$$ in its kernel, here throughout this investigation, \((\gamma )_n \) being the familiar Pochhammer symbol or the shifted factorial, and/or fractional calculus (i.e., differentiation and integration of an arbitrary real or complex order) are presented, for a function f(z), by the familiar differ-integral operator \(_c\mathcal {D}_z^{\mu }[\cdot ],\) defined by $$_c\mathcal {D}_z^{\mu }\big [f(z)\big ]:= \left\{ \begin{aligned} \frac{1}{\Gamma (-\mu )} \int _c^z \frac{f(\tau )}{(z-\tau )^{1-\mu }} d\tau \&\ \big (c\in \mathbb {R}; \mathfrak {R}e(\mu )<0\big ) \\ \frac{d^m}{dz^m}\Big ( {_c\mathcal {D}}_z^{\mu -m}\big [f(z)\big ]\Big ) \&\ \big (m-1\le \mathfrak {R}e(\mu )<m ; m\in \mathbb {N} \big ), \end{aligned} \right. $$ provided that the integral exists, are first established and several consequences of our results are then pointed out.