Schwarz lemma on polydiscs endowed with holomorphic invariant K\"ahler-Berwald metrics

In this paper, we obtain a Schwarz lemma for holomorphic mappings from the unit polydisc $P_m$ into the unit polydisc $P_n$, here $P_m$ and $P_n$ are endowed with $\mbox{Aut}(P_m)$-invariant K\"ahelr-Berwald metric $F_{t,k}$ and $\mbox{Aut}(P_n)$-invariant K\"ahler-Berwald metric $\tilde{F}_{\tilde{t},\tilde{k}}$ respectively. Our result generalizes the Schwarz lemma for holomorphic mappings from $P_m$ into $P_n$ whenever $P_m$ and $P_n$ are endowed with the Bergman metrics respectively. We also obtain a distortion theorem on the unit polydisc $P_m$, where $P_m$ is endowd with an $\mbox{Aut}(P_m)$-invariant K\"ahler-Berwald metric $F_{t,k}$, and show that for each fixed $t\in[0,+\infty)$ and integer $k\geq 2$, $F_{t,k}$ is actually a K\"ahler Finsler-Einstein metric in the sense of T. Aikou.