Noncommutative martingale inequalities associated with convex functions

TL;DR

This paper reviews recent progress in noncommutative martingale inequalities linked with convex functions, highlighting new inequalities, their derivations, and open problems in the field.

Contribution

It consolidates recent advances in noncommutative martingale inequalities involving convex functions, including new results and open questions.

Findings

Noncommutative Burkholder-Gundy inequalities with convex functions

Noncommutative maximal inequalities with convex functions

Open problems in noncommutative martingale theory

Abstract

We report recent advances on noncommutative martingale inequalities associated with convex functions. These include noncommutative Burkholder-Gundy inequalities associated with convex functions due to the present authors and Dirksen and Ricard, noncommutative maximal inequalities associated with convex functions due to Os\c{e}kowski and the present authors, and noncommutative Burkholder and Junge-Xu inequalities associated with convex functions due to Randrianantoanina and Lian Wu. Some open problems for noncommutative martingales are also included.