Sharp and general bounds for Hermite–Hadamard type inequalities via Jensen’s inequality in q-calculus

This paper establishes sharp bounds for Hermite–Hadamard inequalities within the framework of q-calculus by employing q-integrals. To achieve this, the Jensen–Mercer inequality, a generalization of Jensen’s inequality, is utilized with multiple points to derive new and more precise bounds for q-Hermite– Hadamard inequalities. Previous results in classical calculus focused on convex functions and were limited to two points in Jensen’s inequality. By extending the analysis to general points, this work broadens the applicability of these inequalities. The inclusion of left and right q-integrals presents challenges due to the generalized values in the Jensen–Mercer inequality, which are addressed by dividing the analysis into distinct cases. The ability to refine bounds using general points in Jensen–Mercer inequality is a significant outcome, as it unifies and extends many classical results by taking the limit as q → 1− . Numerical examples highlight the effectiveness of this approach, demonstrating that utilizing more points in Jensen–Mercer inequality produces sharper bounds for different values of q-parameter lies in (0, 1) .