Let B be a Banach space,Φ1,Φ2 be two generalized convex Φ-functions and Ψ1,Ψ2 the Young complementary functions of Φl,Φ2 respectively with ∫tt0 ψ2(s)/sds≤c0ψ1(c0t) (t＞t0)for some constants c0 ＞ 0 and t0 ＞ 0,where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 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 respectively; (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)*‖Φl ≤ c‖f*‖Φ2.