Fractional Bullen-Mercer Type Inequalities via Majorization: A Conticrete Approach with Numerical Validation and Applications

Abstract

This article establishes new Bullen-Mercer type inequalities through a unified framework combining majorization theory, convexity, and Riemann-Liouville fractional integrals. A generalized conticrete integral identity is first constructed, which act as the foundation for deriving estimates of the Bullen-Mercer gap in conticrete settings. These estimates are obtained under the convexity assumptions of ||, ||^q (1 < q) , utilizing properties of the modulus, Hölder, power mean, and Young's inequalities. Each theoretical result is accompanied by numerical validation tables for varying fractional orders, as well as 2-D and 3-D graphical illustrations that confirm their validity. Remarks accompanying each result highlight new and previously known inequalities as special cases of the major findings. To demonstrate practical relevance, the main findings are applied to modified Bessel functions of the first kind and to special means, with supporting 2-D visualizations. In last, the work is summed up in a thorough conclusion.