Structural properties and estimation of bounds for Jensen gaps and Hermite–Hadamard inequalities associated with $\sigma$-convex functions

Generalized convexity plays a prolific role in the development and refinement of inequalities. This manuscript assesses some key regularity properties of $\sigma$ convex functions and related generalized fundamental inequalities. Particularly, we proved the relation between $\sigma$-convex functions and classical convexity. Based on this result, we discussed the first-order characterization of $\sigma$-convex functions. Next, we derived some new refinements of Jensen's inequality, including an interpolating Jensen inequality and bounds of Jensen's functional. By taking the benefit of $\sigma$-convexity, we constructed general a Hermite–Hadamard inequality and its weighted analog for symmetric functions as well as and several other inequalities. Our results are novel due to their unifying characteristics because several robust estimates and inequalities can be recaptured for different choices of the transformation function $\sigma$. Finally, we addressed some useful applications to normed spaces, entropy measures, and the general mean inequality.