Affıne Jensen–Mercer, Midpoint, and Trapezoidal Inequalities for Coordinated Convex Functions with Applications to Composite Integral Estimates

Motivated by a recent contribution published in the International Journal of Theoretical Physics, we develop a novel affine Jensen–Mercer inequality for coordinated convex functions that substantially broadens the scope of the existing theory. Building upon this unified framework, we establish new midpoint- and trapezoidal-type inequalities together with their Hölder- and power-mean refinements, thereby extending, generalizing, and complementing results in the aforementioned article. To investigate the influence of the affine structure, we conduct a comprehensive numerical study of the affine parameter c, illustrating its effect on the associated functionals and their corresponding theoretical upper bounds. Furthermore, by applying the proposed inequalities over partitioned rectangles, we derive composite affine midpoint- and trapezoidal-type estimates under coordinated convexity assumptions. Owing to the fundamental role of integral estimation in mathematical physics, these results may also be relevant to the estimation of integral quantities encountered in theoretical physics. Numerical computations and graphical illustrations corroborate the theoretical findings and illustrate the parameter-dependent behavior and flexibility of the proposed affine framework.