In this paper, we give a characterization of Aut(Pn)\documentclass[12pt]{minimal}
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\begin{document}$$\text{ Aut }(P_n)$$\end{document}-invariant strongly pseudoconvex complex Finsler metrics on the unit polydisk in Cn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^n$$\end{document}. We prove that every Aut(Pn)\documentclass[12pt]{minimal}
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\begin{document}$$\text{ Aut }(P_n)$$\end{document}-invariant strongly pseudoconvex complex Finsler metric F on Pn\documentclass[12pt]{minimal}
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\begin{document}$$P_n$$\end{document} is necessary a Kähler–Berwald metric. The holomorphic sectional curvature of (Pn,F)\documentclass[12pt]{minimal}
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\begin{document}$$(P_n,F)$$\end{document} is bounded below and above by two negative constants -K1\documentclass[12pt]{minimal}
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\begin{document}$$-K_1$$\end{document} and -K2\documentclass[12pt]{minimal}
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\begin{document}$$-K_2$$\end{document}, respectively, and the holomorphic bisectional curvature of (Pn,F)\documentclass[12pt]{minimal}
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\begin{document}$$(P_n,F)$$\end{document} is nonpositive and bounded below by a negative constant. This important property makes it possible for us to establish a Schwarz lemma for holomorphic mappings from the unit polydisk Pm\documentclass[12pt]{minimal}
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\begin{document}$$P_m$$\end{document} into Pn\documentclass[12pt]{minimal}
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\begin{document}$$P_n$$\end{document} whenever the polydisks are endowed with holomorphic invariant Kähler–Berwald metrics which are not necessary Hermitian quadratic. As an application, we show that the Schwarz constant of (Pn,F)\documentclass[12pt]{minimal}
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\begin{document}$$(P_n,F)$$\end{document} is both an analytic invariant and a geometric invariant.
