On an Inequality of H. G. Hardy

Abstract

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.

Vectorial fractional integral inequalities with convexity

“…with > , = 1 N We also define the right mixed Riemann-Liouville fractional multiple integral of order α (see also [16]):…”

“…We start with some facts about fractional derivatives needed in the sequel, for more details see, for instance [1,14], and in particular here the next comes from [16]. Let …”

“…He did not write down the constant, but the calculation of the constant was hidden inside his proof. Next we are motivated by [16].…”

On refined Hardy-type inequalities with fractional integrals and fractional derivatives

“…Many mathematicians gave generalizations and improvements of Hardy-type inequalities by giving applications for fractional integrals and fractional derivatives. They discover important and useful Hardy-type inequalities for convex functions as well as for superquadratic functions, (see [4], [6], [9], [11], [12], [13], [16]). In this paper, we obtain some more general Hardy-type inequalities for different kinds of fractional integrals and fractional derivatives like RiemannLiouville fractional integrals, Caputo fractional derivative, fractional integral of a function with respect to an increasing function, Erdelyi-Kóber fractional integrals and Hadamard-type fractional integrals.…”

Improvement of an inequality of G. H. Hardy

“…The calculation for the constant K is hidden inside the proof. In this paper, we give improvements of Hardy type inequalities given in [8]. We also establish new inequalities involving fractional integrals and fractional derivatives of RiemmanLiouville, Caputo, Canavati, Erdelyi-Kóber and Hadamard-type.…”