From Clarkson-McCarthy inequality to Ball-Carlen-Lieb inequality

TL;DR

This paper introduces two new generalizations of Clarkson-McCarthy inequalities involving multiple operators, utilizing advanced mathematical techniques, and completes the proof of an optimal convexity inequality, advancing the theoretical understanding of operator inequalities.

Contribution

It presents two novel generalizations of Clarkson-McCarthy inequalities and completes the proof of an optimal 2-uniform convexity inequality, expanding the theoretical framework of operator inequalities.

Findings

Two new generalizations of Clarkson-McCarthy inequalities

Completion of the proof for the optimal 2-uniform convexity inequality

Presentation of open problems in the field

Abstract

In this paper, we give two new generalizations of Clarkson-McCarthy with several operators, which depends on the unitary orbit technique developed by Bourin, Hadamard Three-lines Theorem and the duality argument developed by Ball, Carlen and Lieb. Moreover, we complete the optimal 2-uniform convexity inequality established by Ball, Carlen and Lieb in [Invent. Math. 115 (1994) 463-482.]. Some open problems are presented.