New Results of Non-Conformable Hermite-Hadamard-Mercer-Type Inequalities Pertaining to Harmonically Convexity Property via Support Line and Their Applications

Abstract

This paper is devoted to the development of new Non-Conformable Hermite–Hadamard–Mercer-type inequalities for harmonically convex functions by means of Non-Conformable fractional integrals. The results provide a natural extension of the classical Hermite–Hadamard-Mercer inequalities, where the framework of harmonic convexity allows us to capture a wider class of functions than standard convexity. In particular, we establish fractional inequalities involving Non-Conformable Hermite–Hadamard–Mercer type expressions for differentiable functions whose derivatives in absolute value are harmonically convex. Corollaries and special cases are derived to demonstrate how the proposed inequalities unify and generalize many earlier results in the literature. Applications are also discussed in the context of inequalities for special means of positive real numbers and q-digamma function showing the practical significance of the theoretical findings. The novelty of this work lies in combining Non-Conformable fractional integrals with Mercer-type refinements, which not only sharpens existing inequalities but also opens a new direction in the study of convexity and fractional analysis. The results obtained here are expected to stimulate further investigations in the areas of functional inequalities, numerical integration, approximation theory, and applied fractional calculus.

Keywords

• Hermite–Hadamard inequality
• Jensen-Mercer inequality
• Non-Conformable fractional integral operators
• Harmonically convex function
• H¨older’s inequality
• Power mean inequality
• q-digamma function
• Special means

Ethical Statement

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. No data were used to support this study. The authors declare that they have no competing interests.

Thanks

The authors wish to thank the editors and reviewers for their valuable comments and suggestions for the betterment of this article.

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