Let B be a Banach space, Φ1, Φ2 be two generalized convex Φ-functions and ψ1, ψ2 the Young complementary functions of Φ1, Φ2 respectively with
∫ tt0ψ2(s)/s ds ≤ c0 ψ1(c0t) (t > t0)
for some constants c0 > 0 and t0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of ψ1 and ψ2, respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (fn)n≥0,
||f *||Φ1 ≤ c||S(p)(f)||Φ2 (or ||S(q)(f)||Φ1 ≤ c||f *||Φ2 , respectively),
where f * and S(p)(f) are the maximal function and the p-variation function of f respec-tively; (ii) If B is a UMD space, Tvf is the martingale transform of f with respect to v = (vn)n≥0 (v* ≤ 1), then ||(Tvf )||Φ1 ≤ c||f *||Φ2 .
