HERMITE-HADAMARD TYPE INEQUALITIES FOR CO-ORDINATED CONVEX FUNCTIONS WITH VARIABLE-ORDER FRACTIONAL INTEGRALS

Abstract

. This study develops a novel framework for Hermite-Hadamard-type inequalities by employing multivariate variable-order Riemann-Liouville fractional integral operators. These operators, which extend classical fractional calculus, allow fractional orders to vary dynamically, providing a powerful tool for capturing spatially and temporally dependent behaviors in multidimensional systems. We rigorously define the variable-order fractional integrals with new formulations of lower and upper bounds tailored for Hermite-Hadamard-type inequalities. By analyzing the properties and well-posedness of the proposed operators, we establish generalized Hermite-Hadamard inequalities for coordinated convex functions. These results represent a significant advancement in fractional analysis, bridging the gap between classical results and the more flexible, dynamic nature of variable-order systems. This work lays the foundation for further exploration of fractional inequalities and their application in systems governed by varying memory effect.