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

Complete complex Finsler metrics and uniform equivalence of the Kobayashi metric