  We prove several noncommutative maximal inequalities associated with convex
functions, including a Doob type inequality for a convex function of maximal
operators on noncommutative martingales, noncommutative Dunford-Schwartz and
Stein maximal ergodic inequalities for a convex function of positive and
symmetric positive contractions. The key ingredient in our proofs is a
Marcinkiewicz type interpolation theorem for a convex function of maximal
operators in the noncommutative setting, which we establish in this paper.
These generalize the results of Junge and Xu in the $L^p$ case to the case of
convex functions.
