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In this paper, we obtain a Schwarz lemma for holomorphic mappings from theunit polydisc $P_m$ into the unit polydisc $P_n$, here $P_m$ and $P_n$ areendowed 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 theSchwarz 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 adistortion 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 thatfor each fixed $t\in[0,+\infty)$ and integer $k\geq 2$, $F_{t,k}$ is actually aK\"ahler Finsler-Einstein metric in the sense of T. Aikou.

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