New Study on the Quantum Midpoint-Type Inequalities for Twice q-Differentiable Functions via the Jensen–Mercer Inequality

The objective of this study is to identify novel quantum midpoint-type inequalities for twice q-differentiable functions by utilizing Mercer’s approach. We introduce a new auxiliary variant the Mercer identity related functions. By applying theory convex identity, we bounds using well-known inequalities, such as H"older’s inequality and power-mean inequality. provide explicit examples along with graphical demonstrations. findings explain previous studies on inequalities. Analytic type, well strategies, have applications in various fields where symmetry plays an important role.