Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 77-91.

WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING FRACTIONAL CALCULUS OPERATORS

Sajid Iqbal, Josip Pečarić, Muhammad Samraiz and Živorad Tomovski

Department of Mathematics, University of Sargodha, Sub-Campus Mianwali, Mianwali, Pakistan
e-mail: sajid_uos2000@yahoo.com, dr.sajid@uos.edu.pk

Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia
e-mail: pecaric@element.hr

Department of Mathematics, University of Sargodha, Sargodha, Pakistan
e-mail: msamraiz@uos.edu.pk

Faculty of Mathematics and Natural Sciences, Gazi Baba bb, 1000 Skopje, Macedonia
e-mail: tomovski@pmf.ukim.mk


Abstract.   The aim of this paper is to give a new class of general weighted Hardy-type inequalities involving an arbitrary convex function with some applications of generalized fractional calculus convolutive operators which contain Gauss-hypergeometric function, generalized Mittag-Leffler function and Hilfer fractional derivative operator, in the kernel.

2010 Mathematics Subject Classification.   26D15, 26D10, 26A33, 34B27.

Key words and phrases.   Inequalities, convex function, fractional derivatives, generalized fractional integral operator.


Full text (PDF) (free access)

DOI: http://doi.org/10.21857/ydkx2cr509


References:

  1. A. Čižmešija, K. Krulić and J. Pečarić, On a new class of refined discrete Hardy-type inequalities, Banach J. Math. Anal. 4 (2010), 122-145.
    MathSciNet     CrossRef

  2. A. Čižmešija, K. Krulić and J. Pečarić, Some new refined Hardy-type inequalities with kernels, J. Math. Inequal. 4 (2010), 481-503.
    MathSciNet     CrossRef

  3. A. Čižmešija, K. Krulić and J. Pečarić, A new class of general refined Hardy type inequalities with kernels, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 17 (2013), 53–80.
    MathSciNet

  4. L. Curiel and L. Galue, A generalization of the integral operators involving the Gauss hypergeometric function, Rev. Técn. Fac. Ingr. Univ. Zulia 19 (1996), 17–22.
    MathSciNet

  5. N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences, Acta Math. Sin. (Engl. Ser.) 27 (2011), 671–684.
    MathSciNet     CrossRef

  6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Elsevier Inc., Amsterdam, 2007.
    MathSciNet

  7. R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for solution of fractional differential equations with generalized Riemann-Liouville fractional derivative, Fract. Calc. Appl. Anal. 12 (2009), 299–318.
    MathSciNet

  8. S. Iqbal, K. Krulić Himmelreich and J. Pečarić, A new class of Hardy-type integral inequalities, Math. Balkanica (N. S.) 28, Fasc. 1-2, (2014), 3-16.

  9. S. Iqbal, K. Krulić Himmelreich and J. Pečarić, On a new class of Hardy-type inequalities with fractional integrals and fractional derivatives, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014) 91–105.
    MathSciNet

  10. S. Iqbal, J. Pečarić, M. Samraiz and Ž. Tomovski, Hardy-type inequalities for generalized fractional integral operators, Tbil. Math. J. 10 (2017), 75–90.
    MathSciNet     CrossRef

  11. S. Kaijser, L. Nikolova, L.-E. Persson and A. Wedestig, Hardy type inequalities via convexity, Math. Inequal. Appl. 8 (2005), 403–417.
    MathSciNet     CrossRef

  12. K. Krulić, J. Pečarić and L.-E. Persson, Some new Hardy-type inequalities with general kernels, Math. Inequal. Appl. 12 (2009), 473–485.
    MathSciNet     CrossRef

  13. K. Krulić, J. Pečarić and D. Pokaz, Inequalities of Hardy and Jensen, Zagreb, Element, 2013.
    MathSciNet

  14. G. M. Mittag-Leffler, Sur la nouvelle fonction, C. R. Acad. Sci. Paris. 137 (1903), 554–558.

  15. T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15.
    MathSciNet

  16. T. O. Salim, Some properties relating to the generalized Mittag-Leffler function, Adv. Appl. Math. Anal. 4 (2009), 21–30.

  17. T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. 5 (2012), 1-13.

  18. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007), 797–811.
    MathSciNet     CrossRef

  19. H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput. 211 (2009), 198–210.
    MathSciNet     CrossRef

  20. Ž. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and Operational Calculus with Generalized Fractional Derivative Operators and Mittag-Leffler Type Functions, Integral Transforms Spec. Funct. 21 (2010), 797-814.
    MathSciNet     CrossRef

  21. A. Wiman, Über den fundamentalsatz in der theorie der functionen, Acta Math. 29 (1905), 191–201.
    MathSciNet     CrossRef


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