  In this paper, we first establish several theorems about the estimation of
distance function on real and strongly convex complex Finsler manifolds and
then obtain a Schwarz lemma from a strongly convex weakly K\"ahler-Finsler
manifold into a strongly pseudoconvex complex Finsler manifold. As
applications, we prove that a holomorphic mapping from a strongly convex weakly
K\"ahler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold
is necessary constant under an extra condition. In particular, we prove that a
holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex
complex Finsler manifold such that its holomorphic sectional curvature is
bounded from above by a negative constant is necessary constant.
