Exponential (h,m)-Convex Functions, Basic Results and Hermite-Hadamard Inequality

This paper explores the extension of the Hermite-Hadamard inequality to exponential (h,m)-convex functions, particularly within the framework of Caputo fractional integrals. Traditional calculus often falls short in adequately modeling systems with memory and non-local interactions, which are prevalent in various scientific and engineering fields. By incorporating Caputo fractional calculus, this work addresses complex dynamic systems that exhibit memory effects, a common characteristic in materials science, financial mathematics, and thermal physics. We present a series of new theoretical results including basic properties and integral inequalities of exponential (h,m)-convex functions, alongside their fractional counterparts. Further, we provide rigorous proofs of the Hermite-Hadamard inequality in both classical and fractional settings, demonstrating its utility in estimating bounds for real-world applications. The paper concludes with a detailed discussion on the practical implications of these findings in optimizing financial models, designing advanced materials, and engineering efficient thermal systems. Our results not only extend the classical understanding of convexity and its applications but also pave the way for future research in fractional calculus and its integration into applied mathematics.