On the maximal inequalities for martingales involving two functions

Let Phi (t) and Psi (t) be nonnegative convex functions, and let phi and psi be the right continuous derivatives of Phi and Psi, respectively. In this paper, we prove the equivalence of the following three conditions: (i) parallel tof*parallel to (Phi) less than or equal to c parallel tof parallel to (Psi), (ii) L-Psi subset of or equal to H-Phi and (iii) integral (t)(s0) phi (s)/s ds less than or equal to c psi (ct), For Allt >s(0), where L-Psi and H-Phi are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.