  In this paper, first of all, according to Lu's and Zhang's works about the
curvature of the Bergman metric on a bounded domain and the properties of the
squeezing functions, we obtain that Bergman curvature of the Bergman metric on
a bounded strictly pseudoconvex domain with $C^2$-boundary or bounded convex
domain is bounded. Secondly, by the property of curvature symmetry on a
K\"ahler manifold, we have the property: if holomorphic sectional curvature of
a K\"ahler manifold is bounded, we can deduce that its sectional curvature is
bounded. After that, applying to the Schwarz lemma from a complete K\"ahler
manifold into a complex Finsler manifold, we get that a bounded strictly
pseudoconvex domain with $C^2$-boundary or bounded convex domain admit complete
strongly pseudoconvex complex Finsler metrics such that their holomorphic
sectional curvature is bounded from above by a negative constant. Finally, by
the Schwarz lemma from a complete K\"ahler manifold into a complex Finsler
manifold, we prove the uniform equivalences of the Kobayashi metric and
Carath\'eodory metric on a bounded strongly convex domain with smoothly
boundary.
