Hybrid harmonic integral inequalities via multiplicative calculus with applications

In this study, we employ proportional Caputo-Hybrid (PCH) operators to establish Hermite-Hadamard (HH) type inequalities for multiplicative harmonically convex functions. A key advantage of these fractional operators lies in their flexibility, allowing the recovery of various forms of inequalities. Specifically, traditional HH-type inequalities for multiplicative harmonically convex functions emerge when the parameter α0 is set to 1, while for multiplicatively differentiable harmonic convex functions, they appear whoen α = 0. To support our findings, we present graphical illustrations based on concrete examples. Additionally, we explore applications to special functions, leading to novel multiplicative fractional order recurrence relations. A promising avenue for future research involves extending these inequalities to interval calculus, where functions take interval values rather than precise numbers, broadening their applicability to uncertainty analysis, numerical approximations, and fractional differential equations.