  We consider some second order quasilinear partial differential inequalities
for real valued functions on the unit ball and find conditions under which
there is a lower bound for the supremum of nonnegative solutions that do not
vanish at the origin. As a consequence, for complex valued functions $f(z)$
satisfying $\partial f/\partial\bar z=|f|^\alpha$, $0<\alpha<1$, and
$f(0)\ne0$, there is also a lower bound for $\sup|f|$ on the unit disk. For
each $\alpha$, we construct a manifold with an $\alpha$-H\"older continuous
almost complex structure where the Kobayashi-Royden pseudonorm is not upper
semicontinuous.
