If $X_0, X_1,\ldots$ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if $V(X_0,X_1,\ldots)$ is the supremum, over stop rules $t$, of $EX_t$, then the set of ordered pairs $\{(x, y): x = V(X_0, X_1,\ldots, X_n)$ and $y = E(\max_{j\leqslant n}X_j)$ for some $X_0,\ldots, X_n\}$ is precisely the set $$C_n = \{(x, y): x \leqslant y \leqslant x\big(1 + n(1 - x^{1/n})\big); 0 \leqslant x \leqslant 1\};$$ and the set of ordered pairs $\{(x, y): x = V(X_0,X_1,\ldots)$ and $y = E(\sup_n X_n)$ for some $X_0, X_1,\ldots\}$ is precisely the set $$C = \bigcup_{n = 1}^\infty C_n$$. As a special case, if $X_0, X_1,\ldots$ is a martingale with $EX_0 = x$, then $E(\max_{j\leqslant n} X) \leqslant x + nx(1 - x^{1/n})$ and $E(\sup_n X_n) \leqslant x - x\ln x$, and both inequalities are sharp.
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