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

Chapter
Part of the Trends in Mathematics book series (TM)

## 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.

## 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

1. 1.
O. Altıntaş, H. Irmak, H.M. Srivastava, Fractional calculus and certain starlike functions with negative coefficients. Comput. Math. Appl. 30(2), 9–15 (1995)
2. 2.
M.P. Chen, H. Irmak, H.M. Srivastava, A certain subclass of analytic functions involving operators of fractional calculus. Comput. Math. Appl. 35(2), 83–91 (1998)
3. 3.
R. Gorenflo, A.A. Kilbas, S.V. Rogosin, On the generalized Mittag-Leffler type functions. Integr. Transform. Spec. Funct. 7, 215–224 (1998)
4. 4.
R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics. Springer Series on CSM Courses and Lectures, vol. 378, ed. by A. Carpinteri, F. Mainardi (Springer, Wien, 1997), pp. 223–276
5. 5.
H.J. Haubold, A.M. Mathai R.K. Saxena, Mittag Leffler functions and their applications, fractional calculus and their consequences. J. Appl. Math., Article ID 298628, 51 pages (2011). http://dx.doi.org/10.11155/2011/298628
6. 6.
R. Hilfer, H. Seybold, Computation of the generalized Mittag-Leffler function and its inverse in the complex plane. Integr. Transform. Spec. Funct. 17, 637–652 (2006)
7. 7.
H. Irmak, B.A. Frasin, A few complex equations constituted by an operator consisting of fractional calculus and their consequences. Chin. J. Math., Article ID 718389, 4 pages (2014). http://dx.doi.org/ 10.1155/2014/718389
8. 8.
H. Irmak, P. Agarwal, Some comprehensive inequalities consisting of Mittag-Leffler type functions in the Complex Plane. Math. Model. Nat. Phenom. 12(3), 65–71 (2017)
9. 9.
I.S. Jack, Functions starlike and convex of order $$\alpha$$. J. Lond. Math. Soc. 3, 469–474 (1971)
10. 10.
A.A. Kilbas, M. Saigo, On Mittag-Leffler type functions, fractional calculus operator and solutions of integral equations. Integr. Transform. Spec. Funct. 4, 355–370 (1996)
11. 11.
A.A. Kilbas, M. Saigo, R.K. Saxena, Solution of Volterra integro-differential equations with generalized MittagLeffler functions in the kernels. J. Integr. Equ. Appl. 14, 377–396 (2002)
12. 12.
A.A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr. Transform. Spec. Funct. 15, 31–49 (2004)
13. 13.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. NorthHolland Mathematical Studies, vol. 204 (Elsevier (North-Holland) Science Publishers, Amsterdam, 2006)Google Scholar
14. 14.
S.S. Miller, P.T. Mocanu, Second-order differential inequalities in the complex plane. J. Math. Anal. Appl. 65, 289–305 (1978)
15. 15.
M. Nunokawa, On properties of non-Caratheodory functions. Proc. Jpn. Acad. Ser. A Math. Sci. 68, 152–153 (1992)
16. 16.
T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohoma Math. J. 19, 7–15 (1971)
17. 17.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Yverdon, 1993)
18. 18.
M. Saigo, A.A. Kilbas, On Mittag-Leffler type function and applications. Integr. Transform. Spec. Funct. 7, 97–112 (1998)
19. 19.
H.J. Seybold, R. Hilfer, Numerical results for the generalized Mittag-Leffler function. Fract. Calc. Appl. Anal. 8, 127–139 (2005)
20. 20.
A.K. Shukla, J.C. Prajabati, On a generalization of MittagLeffler function and its properties. J. Math. Anal. Appl. 336, 797–811 (2007)
21. 21.
H.M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211, 198–210 (2009)
22. 22.
H.M. Srivastava, R.K. Saxena, Some Voterra-type fractional integro-differential equations with a multivariable confluent hypergeometric function as their kernel. J. Integr. Equ. Appl. 17, 199–217 (2005)
23. 23.
Z. Tomovski, T.K. Pogany, H.M. Srivastava, Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity. J. Franklin Inst. 351(12), 5437–5454 (2014)
24. 24.
Z. Tomovoski, R. Hilfer, H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integr. Transform. Spec. Funct. 21(11), 797–814 (2010)