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 ∂f/∂z̄ = |f |α, 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.