Tbilisi Mathematical Journal

Hardy-type inequalities for generalized fractional integral operators

Sajid Iqbal, Josip Pečarić, Muhammad Samraiz, and Živorad Tomovski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The aim of this research paper is to establish the Hardy-type inequalities for Hilfer fractional derivative and generalized fractional integral involving Mittag-Leffer function in its kernel using convex and increasing functions.

Article information

Tbilisi Math. J., Volume 10, Issue 1 (2017), 75-90.

Received: 11 May 2015
Accepted: 14 June 2016
First available in Project Euclid: 26 May 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 26A33: Fractional derivatives and integrals

Hilfer fractional derivative Mittag-Leffer function Fractional integral


Iqbal, Sajid; Pečarić, Josip; Samraiz, Muhammad; Tomovski, Živorad. Hardy-type inequalities for generalized fractional integral operators. Tbilisi Math. J. 10 (2017), no. 1, 75--90. doi:10.1515/tmj-2017-0005. https://projecteuclid.org/euclid.tbilisi/1527300018

Export citation


  • Hardy GH. Notes on some points in the integral calculus, LX. An inequality between integrals. Messenger of Math. 1925;54:150-156.
  • Abramovich S, Kruli´c K, Pečarić J, Persson LE. Some new refined Hardy type inequalities with general kernels and measures. Aequationes Math. 2010;79:157-172.
  • Čižmešija A, Kruli\' c K, Pečarić J. Some new refined Hardy-type inequalities with kernels. J Math Inequal. 2010;4(4):481-503.
  • Čižmešija A, Kruli\' c K, Pečarić J. A new class of general refined Hardy-type inequality with kernels. Rad HAZU. 2013;17:53-80.
  • Farid G, Kruli´c K, Pe¡cari´c J. On refinement of Hardy type inequalities via superquadratic functions. Sarajevo J Math. 2011;7(20):163-175.
  • Iqbal S, Kruli\' c K, Pečarić J. On an inequality of H. G. Hardy. J Inequal Appl. Volume 2010:Article ID 264347.
  • Kruli\' c K, Pečarić J, Persson LE. Some new Hardy type inequalities with general kernels. Math Inequal Appl. 2009;12:473-485.
  • Niculescu C, Persson LE. Convex functions and their applications. A contemporary approach. CMC Books in Mathematics: Springer, New York; 2006.
  • Iqbal S, Kruli\' c K, Pečarić J. On an inequality for convex function with some applications on fractional derivatives and fractional integrals. J Math Inequal. 2011;5(2):219-230.
  • Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204: Elsevier. New York-London; 2006.
  • Tomovski Ž, Hilfer R, Srivastava HM. Fractional and Operational Calculus with Generalized Fractional Derivative Operators and Mittag-Leffler Type Functions. Integral Transforms Spec Funct. 2010;21(11):797-814.
  • Hilfer R, Luchko Y, Tomovski Ž. Operational method for solution of fractional differential equations with generalized Riemann-Liouville fractional derivative. Fractional Calculus & Applied Analysis. 2009;12(3):299-318.
  • Kruli\' c K, Pečarić J, Pokaz D. Inequalities of Hardy and Jensen; 2013.
  • Salim TO, Faraj AW. A generalization of Mittag-Leffler function and integral operator associated with fractional calculus. JFCA. 2012;5:1-13.
  • Salim TO. Some properties relating to the generalized Mittag-Leffler function. Adv Appl Math Anal. 2009;4:21-30.
  • Shukla AK, Prajapati JC. On a generalization of Mittag-Leffler function and its properties. J Math Anal Appl. 2007;336:797-811.
  • Srivastava HM, Tomovski Z. Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal. Appl Math Comput. 2009;211:198-210.
  • Prabhakar TR. A Singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J. 1971;19:7-15.
  • Wiman A. Über den fundamentalsatz in der theorie der Funktionen. Acta Math. 1905;29:191-201.