A version of Hermite-Hadamard-Mercer inequality and associated results

Over the past decade, the Hermite-Hadamard inequality has attracted significant attention from mathematicians, leading to the development of various extensions and generalizations involving different fractional operators, stochastic processes, geometrical interpretations, and applications in image processing. This study focuses on addressing problems related to the Hermite-Hadamard-Jensen-Mercer inequality by incorporating weighted arithmetic means instead of unweighted means within the framework of conformable fractional integral operators. A key objective is to estimate the difference between the derived inequalities. To this end, a novel integral identity is established. Utilizing this identity, several bounds for the difference of the inequalities are obtained by applying convexity, the Hölder inequality, and power-mean inequality. The results are further supported by applications to various means. Finally, the validity and accuracy of the derived results are demonstrated through two-dimensional and three-dimensional graphical representations.