  In this short note, we prove that a complex Finsler vector bundle with
positive Kobayashi curvature must be ample, which partially solves a problem of
S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex
Finsler manifold with positive Kobayashi curvature must be biholomorphic to the
complex projective space; we also show that all Schur polynomials are
numerically positive for complex Finsler vector bundles with positive Kobayashi
curvature.
