Advancements in Harmonic Convexity and Its Role in Modern Mathematical Analysis

Convex functions play an integral part in artificial intelligence by providing mathematical guarantees that make optimization more efficient and reliable. In this manuscript, we originate and analyze a novel category of convexity, namely, harmonically trigonometric p-convex functions, and explore their properties. We provide examples of this new class of convex functions. By leveraging the new convexity, refinements of Hermite–Hadamard-type and Fejér–Hermite–Hadamard-type inequalities are formulated. The derivation of these inequalities involves the utilization of Hölder’s inequality, Hölder–İşcan inequality, the power-mean integral inequality, and certain generalizations associated with these mathematical principles. The validity of the established results is confirmed through visual representation. A comparative analysis is provided to clarify that inequality derived through the power-mean inequality is more refined than other inequalities. Additionally, we discuss the applications of these findings to some special means.