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

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## Abstract

Inequalities in its kernel, here throughout this investigation, \((\gamma )_n \) being the familiar Pochhammer symbol provided that the integral exists, are first established and several consequences of our results are then pointed out.

*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$$E_{\alpha ,\beta }^{\gamma }(z):=\sum _{n=0}^{\infty } \frac{(\gamma )_n}{n!\ \! \Gamma (n\alpha +\beta )}z^n \ \ \ \big ( \beta ,\gamma \in \mathbb {C};\mathfrak {R}e(\alpha )>0\big ),$$

*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$$_c\mathcal {D}_z^{\mu }\big [f(z)\big ]:= \left\{ \begin{aligned} \frac{1}{\Gamma (-\mu )} \int _c^z \frac{f(\tau )}{(z-\tau )^{1-\mu }} d\tau \&\ \big (c\in \mathbb {R}; \mathfrak {R}e(\mu )<0\big ) \\ \frac{d^m}{dz^m}\Big ( {_c\mathcal {D}}_z^{\mu -m}\big [f(z)\big ]\Big ) \&\ \big (m-1\le \mathfrak {R}e(\mu )<m ; m\in \mathbb {N} \big ), \end{aligned} \right. $$

## Keywords

Complex plane Domains in the complex plane Analytic functions Generalized Fractional calculus Equations and inequalities in the complex plane Special functions Mittag-Leffler (type) functions## 2010 Mathematics Subject Classification

30C45 30C50 26A33 26D05 26D15 33E12 33D15 30C80 26E35## References

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