Our goal of this paper is to give a complete characterization of all holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains and establish a corresponding Schwarz lemma for holomorphic mappings with respect to these invariant metrics. We prove that every \({{\text{Aut}}}({\mathfrak {D}})\) -invariant strongly pseudoconvex complex Finsler metric F on a classical domain \({\mathfrak {D}}\) is a Kähler-Berwald metric which is not necessary Hermitian quadratic, but it enjoys very similar curvature property as that of the Bergman metric on \({\mathfrak {D}}\) . In particular, if F is Hermitian quadratic, then F must be a constant multiple of the Bergman metric on \({\mathfrak {D}}\) . This actually answers the 4-th open problem posed by Bland and Kalka (Variations of holomorphic curvature for Kähler Finsler metrics, American Mathematical Society, Providence, 1996). We also obtain a general Schwarz lemma for holomorphic mappings from a classical domain \({\mathfrak {D}}_1\) into another classical domain \({\mathfrak {D}}_2\) whenever \({\mathfrak {D}}_1\) and \({\mathfrak {D}}_2\) are endowed with arbitrary holomorphic invariant Kähler-Berwald metrics \(F_1\) and \(F_2\) , respectively. The method used to prove the Schwarz lemma is purely geometric. Our results show that the Lu constant of \(({\mathfrak {D}},F)\) is both an analytic invariant and a geometric invariant. This can be better understood in the complex Finsler setting.
