Beyond classical convexity: super-quadratic convex mappings and their inequality framework

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 73 - 30
• Main Authors: Javed, Muhammad Zakria, Arshad, Amina, Awan, Muhammad Uzair, Almalki, Yahya, Breaz, Daniel, Cotîrlă, Luminiţa-Ioana, Danciu, Alin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 13.06.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Control theory; Convexity; Hadamard’s inequality; Hermite; Inequality; Information theory; Jensen; Jensen’s inequality; Mathematics; Mathematics and Statistics; Mercer inequality; Operators (mathematics); Optimization; Super-quadratic mapping; Jensen; Jensen’s inequality; Hadamard’s inequality; 26A51; 26D20; 26D10; 26A33; Super-quadratic mapping; Fractional calculus; Hermite; Mercer inequality; 26D07; 26D15
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03325-3

This study investigates a new class of totally ordered interval-valued super-quadratic convex mappings that are based on the ordering relation c r and control mappings ħ . We also discuss a few captivating properties of this class, along with some special cases, to ensure its broadness and utility. We look at classical integral inequalities such as Jensen’s, reverse Jensen’s, Jensen–Mercer, and Hermite–Hadamard’s inequality along with interval-valued c r - h super-quadratic mappings in the context of classical and fractional operators from a useful point of view. The outcomes are affirmed through illustrative examples, visuals, and applications to means and information theory, respectively.