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

Adam Coffman, Yifei Pan

article abstract

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
df/dzbar=|f|^alpha, 0<\alpha<1, and f(0) not =0, there is also a lower
bound for sup|f| on the unit disk.  For each alpha, we construct a
manifold with an alpha-Holder continuous almost complex structure
where the Kobayashi-Royden pseudonorm is not upper semicontinuous.