$k$ -Fractional Variants of Hermite-Mercer-Type Inequalities via $s$-Convexity with Applications

Abstract

This article is aimed at studying novel generalizations of Hermite-Mercer-type inequalities within the Riemann-Liouville k -fractional integral operators by employing s -convex functions. Two new auxiliary results are derived to govern the novel fractional variants of Hadamard-Mercer-type inequalities for differentiable mapping Ψ whose derivatives in the absolute values are convex. Moreover, the results also indicate new lemmas for Ψ ′ , Ψ ′ ′ , and Ψ ′ ′ ′ and new bounds for the Hadamard-Mercer-type inequalities via the well-known Hölder’s inequality. As an application viewpoint, certain estimates in respect of special functions and special means of real numbers are also illustrated to demonstrate the applicability and effectiveness of the suggested scheme.

Fractional Mercer's Hermite–Hadamard type inequalities in the frame of interval analysis and its applications to matrix

“…In above inequality T represent a convex function on closed interval [π 1 , π 2 ]. For the amazing literature regarding above inequality, one can refer [8,36,40,54].…”

Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators

“…Other variants of the Jensen-Mercer inequality (2), for different notions of convexity, can be found in [16,[22][23][24][25].…”

Fejér–Pachpatte–Mercer-Type Inequalities for Harmonically Convex Functions Involving Exponential Function in Kernel

“…Recently (from 2020 to 2021), some new kinds of fractional treatment of Hermite-Jensen-Mercer-type inequalities for a variety of fractional integral operators were presented in [25][26][27]. All these results were investigated for convex functions or s-convex functions, and many applications to special functions like Bessel and q-digamma functions were obtained.…”

“…Replacing a 1 and a 2 by 1/(1/θ 1 ) + (1/θ 2 ) − (1/a 1 ) and 1/(1/θ 1 ) + (1/θ 2 ) − (1/a 2 ), respectively, in (27), we get…”