On the maximal inequalities for martingales involving two functions

Let Phi (t) and Psi (t) be nonnegative convex functions, and let phi and ps i be the right continuous derivatives of Phi and Psi, respectively. In this paper, we prove the equivalence of the following three conditions: (i) par allel tof*parallel to (Phi) less than or equal to c parallel tof parallel t o (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 suffic ient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.