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On an inequality of G. H. Hardy. (English) Zbl 1229.26014

This extensive paper is devoted mainly to the integral inequalities and fractional integral inequalities of different types. Without doubt this paper is worthy of note.
First, a nice general integral inequality for convex and increasing functions is proved and then applied to obtain generalizations of a classical inequality of G. H. Hardy (i.e., on the fact that the Riemann-Liouville fractional integral operators \(I^{\alpha}_{a+}f\) and \(I^{\alpha}_{b-}f\) are bounded in \(L_p(a,b)\), \(1\leq p\leq \infty\)). Refinements of this inequality are also given.
A second group of inequalities presented in the paper is connected with the generalized Riemann-Liouville fractional derivative, the generalized \(\nu\)-fractional derivative of \(f\) over \([a,b]\), the Caputo fractional derivative, the fractional integrals of a function with respect to another function, the Erdélyi-Kober type fractional integrals and radial fractional derivatives – the results are similar to those mentioned above which generalize the Hardy inequality.
Finally, the authors apply again the first and simultaneously the main result of the paper to multidimensional settings in order to obtain new results involving mixed Riemann-Liouville fractional integrals.

MSC:

26A33 Fractional derivatives and integrals
26D15 Inequalities for sums, series and integrals
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[1] Anastassiou GA: Fractional Differentiation Inequalities. Springer Science+Businness Media, LLC, Dordrecht, the Netherlands; 2009:xiv+675. · Zbl 1181.26001 · doi:10.1007/978-0-387-98128-4
[2] Handley GD, Koliha JJ, Pečari ć J: Hilbert-Pachpatte type integral inequalities for fractional derivatives. Fractional Calculus & Applied Analysis 2001, 4(1):37-46. · Zbl 1030.26012
[3] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier, New York, NY, USA; 2006:xvi+523.
[4] Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives : Theory and Applications. Gordon and Breach Science Publishers, Yverdon, Switzerland; 1993:xxxvi+976. · Zbl 0818.26003
[5] Hardy H: Notes on some points in the integral calculus. Messenger of Mathematics 1918, 47(10):145-150.
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