Optimal L${}^1$-bounds for submartingales

The optimal function $f$ satisfying $$\mathbb{E}\mathrm{\mid }\sum _1^n{X}_i\mathrm{\mid }\ge f(\text{mathrbb}E\mathrm{\mid }{X}_1\mathrm{\mid },\mathrm{.}\mathrm{.}\mathrm{.},\mathbb{E}\mathrm{\mid }{X}_n\mathrm{\mid })$$ for every martingale $(X_1,X_1+X_2,\mathrm{.}\mathrm{.}\mathrm{.},{\sum }_{i=1}^n{X}_i)$ is shown to be given by $$f(a)=\max \{a_k-\sum _{i=1}^{k-1}{a}_i{\}}_{k=1}^n\cup \{\frac{a_k}{2}{\}}_{k=3}^n$$ for $a\in {[0,\mathrm{\infty }[}_{}^n$. A similar result is obtained for submartingales $(0,X_1,X_1+X_2,\mathrm{.}\mathrm{.}\mathrm{.},{\sum }_{i=1}^n{X}_i)$. The optimality proofs use a convex-analytic comparison lemma of independent interest.