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

  • Hüseyin IrmakEmail author
  • Praveen Agarwal
Part of the Trends in Mathematics book series (TM)


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.


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 


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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceÇankırı Karatekin UniversityÇankırıTurkey
  2. 2.Department of MathematicsAnand International College of EngineeringJaipurIndia
  3. 3.International Centre for Basic and Applied SciencesJaipurIndia

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