Functional inequalities in the framework of Banach spaces

TL;DR

This paper extends a quadrilateral inequality from Hilbert spaces to Banach spaces using majorization theory and the von Neumann-Jordan constant, leading to new functional inequalities in Banach space geometry.

Contribution

It introduces a generalized quadrilateral inequality in Banach spaces, broadening the scope of classical inequalities through novel theoretical methods.

Findings

Extended quadrilateral inequality to Banach spaces

Derived new functional inequalities in Banach space geometry

Connected inequalities with classical linear algebra results

Abstract

A quadrilateral inequality established by C. Sch\"otz in the context of Hilbert spaces is extended to the framework of Banach spaces. Our approach is based on the majorization theory and a substitute for the parallelogram law associated with Clarkson's notion of von Neumann-Jordan constant. As a by-product, several functional inequalities that extend classical inequalities from linear algebra and geometry of Banach spaces are also obtained.