The Hermite-Hadamard-Mercer Type Inequalities via Generalized Proportional Fractional Integral Concerning Another Function

In order to be able to study cosmic phenomena more accurately and broadly, it was necessary to expand the concept of calculus. In this study, we aim to introduce a new fractional Hermite–Hadamard–Mercer’s inequality and its fractional integral type inequalities. To facilitate that, we use the proportional fractional integral operators of integrable functions with respect to another continuous and strictly increasing function. Moreover, we establish some new fractional weighted φ -proportional fractional integral Hermite–Hadamard–Mercer type inequalities. Furthermore, in this article, we are keen to present some special cases related to our current study compared to the previous work of the inequality under study.