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Let φ1,φ2 be nonnegative nondecreasing functions, and φ1 be concave. Theauthors prove the equivalence of the following two conditions:(i) Eφ1(Mf) 【 cEφ2(Z0+A∞) for every nonnegative submartingale / = (fn)n≥o with it’sDoob’s Decomposition: f = Z + A, where