Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

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 , 0<α<1, and f(0)≠0, there is also a lower bound for sup|f| on the unit disk. For each α, we construct a manifold with an α-Hölder continuous almost complex structure where the Kobayashi–Royden pseudonorm is not upper semicontinuous.