  Let $X$ be a supermartingale starting from $0$ which has only nonnegative
jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and
$\mathfrak{c}_p$ such that $$ \,\,\,\,\sup_{t\geq
0}\left|\left|X_t\right|\right|_p\leq C_p\left|\left|-\inf_{t\geq
0}X_t\right|\right|_p,$$ $$ \,\,||\sup_{t\geq 0}X_t||_p\leq
c_p\left|\left|-\inf_{t\geq 0}X_t\right|\right|_p$$ and $$ ||\sup_{t\geq
0}|X_t|\;||_p\leq \mathfrak{c}_p\left|\left|-\inf_{t\geq
0}X_t\right|\right|_p.$$ The estimates are shown to be sharp if $X$ is assumed
to be a stopped one-dimensional Brownian motion. The inequalities are deduced
from the existence of special functions, enjoying certain majorization and
convexity-type properties. Some applications concerning harmonic functions on
Euclidean domains are indicated.
