Functional Inequalities in the Framework of Banach Spaces

Abstract This paper aims to illustrate the usefulness of majorization theory and higher-order convexity in generalizing several classical inequalities as functional inequalities. This includes results due to Ball et al. (Invent Math 115:463-482, 1994), Hanner (Ark Mat 3:239-244, 1956), Popoviciu (Analele ştiinţifice Univ “Al. I. Cuza” Iasi, Secţia I-a Mat 11:155-164, 1965) and Zhang (Matrix Theory: Basic Results and Techniques, Springer, New York, 2011). Also, based on a substitute for the parallelogram law due to Clarkson, a quadrilateral inequality established by Schötz (in Quadruple inequalities: Between Cauchy-Schwarz and triangle. ArXiv preprint, arXiv:2307.01361 v.3, 2024) in the context of Hilbert spaces is extended to the general framework of Banach spaces.