  We establish a lower estimate for the Kobayashi-Royden infinitesimal
pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly
plurisubharmonic function. We apply this result to study the boundary behaviour
of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient
condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we
obtain the regularity up to the bounday of $J$-holomorphic discs attached to a
totally real submanifold in $M$.
