Some Sharp and General Bounds for q-Hermite--Hadamard Type Inequalities utilizing Jensen's inequality

Abstract

This paper establishes sharp bounds for Hermite--Hadamard inequalitieswithin the framework of $q$-calculus by employing $q$-integrals. To achievethis, the Jensen--Mercer inequality, a generalization of Jensen'sinequality, is utilized with multiple points to derive new and more precisebounds for $q$-Hermite--Hadamard inequalities. Previous results in classicalcalculus focused on convex functions and were limited to two points inJensen's inequality. By extending the analysis to general points, this workbroadens the applicability of these inequalities. The inclusion of left andright $q$-integrals presents challenges due to the generalized values in theJensen--Mercer inequality, which are addressed by dividing the analysis intodistinct cases. The ability to refine bounds using general points inJensen--Mercer inequality is a significant outcome, as it unifies andextends many classical results by taking the limit as $q\rightarrow 1^{-}$.Numerical examples highlight the effectiveness of this approach,demonstrating that utilizing more points in Jensen--Mercer inequalityproduces sharper bounds for different values of $q$-parameter lies in $%\left( 0,1\right) .$