Maximal $l\phi$-inequalities for nonnegative submartingales

Let $(M_n)_{n\ge 0}$ be a nonnegative submartingale and let $M_n^*\stackrel{\textrm{def}}{=}\max_{0\le k\le n}M_k$, $n\ge 0$, be the associated maximal sequence. For nondecreasing convex functions $\phi\colon[0,\infty)\to[0,\infty)$ with $\phi(0)=0$ (Orlicz functions) we provide various inequalities for $E\phi(M_n^*)$ in terms of $E\Phi_a(M_n)$, where, for $a\ge 0$,
$$ \Phi_{a}(x)\,\stackrel{\textrm{def}}{=}\,\int_{a}^{x}\!\!\int_{a}^{s}\frac{\phi'(r)}{r}\,dr\,ds, \qquad x>0. $$
Of particular interest is the case $\phi(x)=x$ for which a variational argument leads us to
$$ EM_n^*\le\Bigg(1+\bigg(E\bigg(\int_{1}^{M_n\vee 1}\log x\,dx\bigg)\bigg)^{1/2}\Bigg)^2. $$
A further discussion shows that the given bound is better than Doob's classical bound $e(e-1)^{-1}(1+\textbf E M_n\log^{+}M_n)$ whenever $\textbf E(M_n-1)^{+}\ge e-2\approx 0.718$.