  In this paper, we consider some generalized holomorphic maps between
pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the
transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type
Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we
are able to establish several Schwarz type results for both the \emph{CR} maps
and the transversally holomorphic maps between pseudo-Hermitian manifolds.
Finally, we also discuss the \emph{CR} hyperbolicity problem for
pseudo-Hermitian manifolds.
