A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains

We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc $${\mathbb {G}}_n$$ , family of domains naturally associated with spectral interpolation, defined by $$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum i<j\le n}z_iz_j \ldots \prod _{i=1}^n z_i \right) : \,|z_i|<1, i=1,\ldots ,n \right\} . \end{aligned}$$ first few extended $$\widetilde{{\mathbb {G}}}_n$$ introduced in [35] and following way: \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {C}}^n :\; q \in {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, {C}} \text { }\\&\quad |\beta _j|+ _{n-j}| < {n \atopwithdelims ()j} } j=1,\ldots n-1 \}. then show that these are provide It is easy verify {G}}_n=\widetilde{{\mathbb $$n=1,2$$ $${{\mathbb {G}}}_n \subsetneq $$n\ge 3$$ As consequence {G}}_n}$$ we have analogous Since point $$(s_1,\ldots s_{n-1},p)\in $${n ()i}$$ least upper bound $$|s_i|$$ which same $$|y_i|$$ any $$(y_1,\ldots ,y_{n-1},q) $$1\le i \le n-1$$ become too. conditions necessary sufficient when $$n=1,2, In particular $$n=2$$ our results add new existing bidisc.