On some Hardy-type inequalities for generalized fractional integrals

. In this article we establish the variant of Hardy-type and re fi ned Hardy-type inequal- ities for a generalized Riemann-Liouville fractional integral operator and Riemann-Liouville k -fractional integral operator using convex and monotone convex functions. We also discuss one dimensional cases of our related results. As special cases of our general results we obtain the consequences of Iqbal et al. [11]. We also obtained exponentially convex linear functionals for the generalized fractional integral operators. Moreover, it includes Cauchy means for the above mentioned operators. The fi rst de fi nition is presented in