Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

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.