We prove that for almost complex structures of H\"older class at least 1/2,
any J-holomorphic disc, that is constant on some non empty open set, is
constant. This is in striking contrast with well known, trivial, non-uniqueness
results. We also investigate uniqueness questions (do vanishing on some open
set, or vanishing to infinite order, or having a non isolated zero, imply
vanishing) in connection with differential inequalities that arise in the
theory of almost complex manifolds. The case of vector valued functions is
different from the case of scalar valued functions
