On the maximal inequalities for martingales involving two functions

Let $\Phi (t)$ and $\Psi (t)$ be nonnegative convex functions, and let $\varphi$ 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) $\Vert f^{*}\Vert _\Phi \leq c\Vert f\Vert _\Psi ,$ (ii) $L^\Psi \subseteq$ $H^\Phi$and (iii) $\int_{s_0}^t\frac{\varphi (s)}sds\leq c\psi (ct), \forall t>s_0,$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.