On the Integral Representation and the Raşa, Jensen and Hermite–Hadamard Inequalities for Box-Convex Functions

Abstract

We give the integral representation of Popoviciu’s (Mathematica 8:1–85, 1934) box-(m, n)-convex functions. Based on this integral representation, we obtain a characterization of box-(m, n)-convex orders, which we then use in the proofs of the Raşa type, the Hermite–Hadamard type and the Jensen inequalities for box-(m, n)-convex functions as well as for strongly box-(m, n)-convex functions. We extend the results obtained by Gal and Niculescu (J Math Anal Appl 479:903–925, 2019), and by Gavrea and Gavrea (Mediterr J Math 19:18–33, 2022).

Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for (h1,h2)–Godunova–Levin Preinvex Function with Applications and Two Open Problems

“…As we connect the results of a generalized class of Godunova-Levin mappings with probability density functions and find bounds with the aid of moment-generating functions, it could be interesting for academics to take inspiration from these results and discuss other properties like variance, standard deviation, and various other applications of moments. Here, we refer to [48][49][50][51][52].…”