Refinements of some fractional integral inequalities involving extended convex functions and fractional Caputo derivatives

This article examines famous fractional Hermite–Hadamard integral inequalities through the applications of fractional Caputo derivatives and extended convex functions. We develop modifications involving two known classical fractional extended versions of the Hermite–Hadamard type inequalities for the convex function extensions. We also obtain the refinements of some inequalities like Fejér–Hadamard, trapezoidal, and midpoint type for two extended convex functions by applying the Caputo fractional derivative identities. The presented results yield both generalizations and improvements upon previously established inequalities.