Advertisement

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

  • Hüseyin Irmak
  • Praveen Agarwal
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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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. 223–276CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    I.S. Jack, Functions starlike and convex of order \(\alpha \). J. Lond. Math. Soc. 3, 469–474 (1971)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Nunokawa, On properties of non-Caratheodory functions. Proc. Jpn. Acad. Ser. A Math. Sci. 68, 152–153 (1992)CrossRefGoogle Scholar
  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)MathSciNetzbMATHGoogle Scholar
  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)zbMATHGoogle Scholar
  18. 18.
    M. Saigo, A.A. Kilbas, On Mittag-Leffler type function and applications. Integr. Transform. Spec. Funct. 7, 97–112 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H.J. Seybold, R. Hilfer, Numerical results for the generalized Mittag-Leffler function. Fract. Calc. Appl. Anal. 8, 127–139 (2005)MathSciNetzbMATHGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetzbMATHGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations