Strengthening of Clarkson-McCarthy inequalities with several operators

TL;DR

This paper strengthens Clarkson-McCarthy inequalities involving multiple operators, confirming a conjecture, improving previous results, and extending to infinite cases with applications to eigenvalue inequalities.

Contribution

The paper introduces new strengthened inequalities for multiple operators, confirming a conjecture and generalizing previous results, including infinite cases and eigenvalue bounds.

Findings

Confirmed a conjecture on Clarkson-McCarthy inequalities.

Improved bounds compared to previous results by Hirazallah-Kittaneh and Bhatia-Kittaneh.

Extended inequalities to infinite operator cases and derived related eigenvalue inequalities.

Abstract

Strengthening of two Clarkson-McCarthy inequalities with several operators is established. These not only confirm a conjecture of the author in [Israel J. Math. 2024], but also improve results of Hirazallah-Kittaneh in [Integral Equations Operator Theory 60 (2008)] and Bhatia-Kittaneh in [Bull. London Math. Soc. 36 (2004)]. We also give a generalization of a result for pairs $(A_i,B_i), 1\le i\le n$ and obtain another form of another inequality. The infinite cases of these inequalities are also discussed. The method is an extension of the result obtained by Bourin and Lee in [Linear Algebra Appl. 601 (2020)]. Some related eigenvalue inequalities are also obtained.