Functional inequalities in the framework of Banach spaces

A quadrilateral inequality established by C. Schötz 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.