Jensen-Mercer and related Inequalities for Coordinated Convex Functions with their Computational Analysis and Applications

Abstract

In this article, we define the Jensen-Mercer inequality for coordinated convex functions. Defining Jensen-Mercer inequality on coordinates was a challenging problem for researchers working in the field of inequalities. Newly established inequality extends Jensen’s classical result to coordinate-wise convex functions, using Mercer-type conditions to derive bounds in multi-variable settings. It is instrumental in optimization and convex analysis for systems with interdependent variables under convex constraints. By use of a new definition, we establish Hermite–Hadamard-Mercer type inequalities for coordinated convex functions. Hermite–Hadamard-Mercer type inequality is the generalization of Hermite-Hadamard inequality, we can get Hermite-Hadamard inequality by taking endpoints in Hermite–Hadamard-Mercer type inequality. Also, we prove Midpoint–Mercer and Trapezoid–Mercer type inequalities on coordinates. The newly established inequalities are valid for the functions that are coordinated convex but may not be convex. Moreover, we give numerical examples to check the validity of newly established results and show that the bounds proved in this paper are better than the previously established results. The results of the study show that the inequalities hold for a wider range of functions. The use of new results can improve the accuracy of modeling and optimization of systems in fields such as economics, engineering, and physics. Also, we provide applications of the numerical integration methods of newly established results to develop the interest of the readers.