Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators

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DOI: 10.46793/KgJMat2105.797I


In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.


Inequalities, convex functions, Riemann-Liouville fractional derivative.


[1]   E. Adeleke, A. Čižmešija, J. Oguntuase, L. E. Persson and D. Pokaz, On a new class of Hardy-type inequalities, J. Inequal. Appl. 259 (2012).

[2]   D. Baleanu, P. Agarwal, R. K. Parmar, M. M. Alquarashi and S. Salahshour, Extension of the fractional derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl. 10 (2017), 2914–2924.

[3]   S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66.

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

[5]   N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences, Acta Math. Sin. (Engl. Ser.) 27(4) (2011), 671–684.

[6]   S. Iqbal, J. Pečarić and Y. Zhou, Generalization of an inequality for integral transforms with kernel and related results, J. Inequal. Appl. 2010 (2010), Article ID 948430.

[7]   S. Iqbal, K. Krulić, J. Pečarić and Dora Pokaz, n-Exponential convexity of Hardy-type and Boas-type functionals, J. Math. Inequal. 7(4) (2011), 739–750.

[8]   S. Iqbal, J. Pečarić, M. Samraiz and N. Sultana, Applications of refined Hardy-type inequalities, Math. Inequal. Appl. 18(4) (2015), 1539–1560.

[9]   S. Iqbal, J. Pečarić, M. Samraiz and Z. Tomovski, Hardy-type inequalities for generalized fractional integral operators, Tbilisi Math. J. 10(1) (2017), 1–16.

[10]   A. A. Kilbas, H. M. Sarivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Sciences B.V., Amsterdam, 2006.

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

[12]   K. Mehrez and Z. Tomovski, On a new (p,q)-Mathieu type power series and its applications, Appl. Anal. Discrete Math. (2019) (to appear).

[13]   C. Niculescu and L. E. Persson, Convex Functions and Their Applications. A Contemporary Approach, CMC Books in Mathematics Springer, New York, 2006.

[14]   M. A. Ozerslan and E. Ozergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 (2010), 1825–1833.

[15]   J. Pečarić and J. Perić, Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39(1) (2012), 65–75.

[16]   L. E. Persson, M. A. Ragusa. N. Samko and P. Wall, Commutators of Hardy operators in vanishing Morrey spaces, American Institute of Physics 1493 (2012), 859–866.

[17]   G. Rahman, S. Mubeen, K. S. Nisar and J. Choi, Certain extended special functions and fractional integral operator via an extended beta function, Nonlinear Functional Analysis and Applications 4(1) (2019), 1–13.

[18]   G. Rahman, K. S. Nisar and Z. Tomovski, A new extention of Riemann-Liouville fractional derivative operator, Commun. Korean Math. Soc. 34(2) (2019), 507–522.

[19]   S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[20]   M. Shadab, S. Jabee and J. Choi, An extension of beta function and its application, Far East J. Math. Sci. 103(1) (2018), 235–251.