Best possible bounds of the von Bahr--Esseen type

The well-known von Bahr--Esseen bound on the absolute $p$th moments of martingales with $p\in(1,2]$ is extended to a large class of moment functions, and now with a best possible constant factor (which depends on the moment function). This result appears to be new even for the power moments. As an application, measure concentration inequalities for separately Lipschitz functions on product spaces are obtained. Relations with $p$-uniformly smooth and $q$-uniformly convex normed spaces are discussed.