Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that N is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings f from M into N. As applications, we obtain a Liouville type rigidity result for holomorphic mappings f from M into N, as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.