Schwarz lemma on polydiscs endowed with holomorphic invariant Kähler-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 $Aut(P_m)$-invariant Kähelr-Berwald metric $F_{t,k}$ and $Aut(P_n)$-invariant Kähler-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 $Aut(P_m)$-invariant Kähler-Berwald metric $F_{t,k}$, and show that for each fixed $t\in [0,+\mathrm{\infty })$ and integer $k\ge 2$, $F_{t,k}$ is actually a Kähler Finsler-Einstein metric in the sense of T. Aikou.