Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences

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Let $\Phi (\cdot)$ be a nondecreasing convex function on $[0, \infty)$. We show that for any integer $n \geq 1$ and real $a$, $$E \Phi ((M_n - a)^+) \leq 2E \Phi ((S_n - a)^+) - \Phi (0)$$ and $$E(M_n \vee \mathrm{med} S_n) \leq E|S_n - \mathrm{med} S_n|.$$ where $X_1, X_2, \dots$ are any independent mean zero random variables with partial sums $S_0 = 0, S_k = X_1 + \dots + X_k$ and partial sum maxima $M_n = \max_{0 \leq k \leq n} S_k$. There are various instances in which these inequalities are best possible for fixed $n$ and/or as $n \to \infty$. These inequalities remain valid if $\{X_k\}$ is a martingale difference sequence such that $E(X_k \{X_i; i \not= k\}) = 0$ a.s. for each $k \geq 1$. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.