Sharp L1(ℓq)L1(ℓq) estimate for a sequence and its predictable projection

Let f=(fn)n≥0f=(fn)n≥0 be a sequence of integrable Banach-space valued random variables and g=(gn)n≥0g=(gn)n≥0 denote its predictable projection. We prove that, for 1≤q<∞1≤q<∞, E(∑n=0∞|gn|q)1/q≤2(q−1)/qE(∑n=0∞|fn|q)1/q and that the constant 2(q−1)/q2(q−1)/q is the best possible. The proof rests on the construction of a certain special function enjoying appropriate majorization and concavity.