Sharp norm comparison of the maxima of a sequence and its predictable projection

Let f=(fn)f=(fn) be an adapted sequence of integrable Banach-space-valued random variables and g=(gn)g=(gn) denote its predictable projection. We prove that, for 1≤p≤∞1≤p≤∞, ‖supn‖gn‖‖p≤(1+(p−1)p−1pp)‖supn‖fn‖‖p and the constant 1+(p−1)p−1pp is the best possible.