In this paper we give upper bounds on the tail or the quantiles of the one-sided maximum of a nonnegative submartingale in the class L log L or the maximum of a submartingale in L p . Our upper bounds involve the entropy in the case of nonnegative martingales in the class L log L and the L p -norm in the case of submartingales in L p . Starting from our results on entropy, we also improve the so-called bounded differences inequality. All the results are based on optimal bounds for the conditional value at risk of real-valued random variables.