Some novel inequalities involving a function's fractional integrals in relation to another function through generalized quasiconvex mappings

Abstract

In this paper, we establish new inequalities of the Hermite--Hadamard, midpoint and trapezoid types for functions whose first derivatives in absolute value are $\eta$-quasiconvex by means of generalized fractional integral operators with respect to another function $\o:[\a,\b]\to(0,\infty)$. Our theorems amount to results involving the Riemann--Liouville fractional integral operators if $\o$ is the identity map, and results involving the Hadamard operators if $\o(x)=\ln x$. More inequalities can be deduced by choosing different bifunctions for $\eta$. To the best of our knowledge, the results obtained herein are new and we hope that they will stimulate further interest in this direction.