Improvement of an inequality of G. H. Hardy

Abstract

Abstract. In this paper, we give an improvement of an inequality of G. H. Hardy using fractional integrals and fractional derivatives. We also obtain means of Cauchy type and prove their monotonicity.

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

“…After introduction, in Section 2 and Section 3, we give more generalized results related to inequalities of G. H. Hardy for different kind of fractional integrals and fractional derivatives. Also we obtain some particular results for inequalities of G. H. Hardy which are discussed in [11] (see also [10], [12]). …”

“…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.…”