Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

TL;DR

This paper extends noncommutative martingale inequalities to convex functions, providing $ ext{Phi}$-moment analogues of classical inequalities for a broad class of Orlicz functions, generalizing prior results.

Contribution

It introduces $ ext{Phi}$-moment versions of noncommutative Burkholder and Rosenthal inequalities for convex Orlicz functions with specific index ranges, broadening the scope of these inequalities.

Findings

Established $ ext{Phi}$-moment inequalities for convex Orlicz functions.

Generalized noncommutative Burkholder inequalities.

Extended noncommutative Rosenthal inequalities.

Abstract

We prove noncommutative martingale inequalities associated with convex functions. More precisely, we obtain $\Phi$-moment analogues of the noncommutative Burkholder inequalities and the noncommutative Rosenthal inequalities for any convex Orlicz function $\Phi$ whose Matuzewska-Orlicz indices $p_\Phi$ and $q_\Phi$ are such that $1<p_\Phi\leq q_\Phi <2$ or $2<p_\Phi \leq q_\Phi<\infty$. These results generalize the noncommutative Burkholder/Rosenthal inequalities due to Junge and Xu.