We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetilde{\Gamma}_n$.

中文翻译：

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两个域族和复杂几何的 Schwarz 引理

我们对对称多圆盘 $\gn$ 和扩展对称多圆盘 $\Gn$ 进行了严格的估计以获得 Schwarz 型引理。我们在特定条件下显式地构造了一个插值函数。为此，我们遵循了 \cite{Young-LMS} 中描述的方法。我们还发现 $\Gn$ 家族的成员与其闭包 $\widetilde{\Gamma}_n$ 之间存在一些几何相互作用。