  We make some sharp estimates to obtain a Schwarz lemma for the
\textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally
associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{
\left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots,
\prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a
few estimates for the \textit{the extended symmetrized polydisc}
$\widetilde{\mathbb G}_n$, a family of domains introduced in \cite{pal-roy 4}
and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n :=
\Bigg\{ (y_1,\dots,y_{n-1}, q)\in \C^n :\; q \in \mathbb D, \; y_j = \be_j +
\bar \be_{n-j} q, \; \beta_j \in \mathbb C &\text{ and }\\ |\beta_j|+
|\beta_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}.
\end{align*} We then show that these estimates are sharp and provide a Schwarz
lemma for $\Gn$. It is easy to verify that $\mathbb G_n=\widetilde{\mathbb
G}_n$ for $n=1,2$ and that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$
for $n\geq 3$. As a consequence of the estimates for $\widetilde{\mathbb G_n}$,
we have analogous estimates for $\mathbb G_n$. Since for a point $(s_1,\dots,
s_{n-1},p)\in \mathbb G_n$, ${n \choose i}$ is the least upper bound for
$|s_i|$, which is same for $|y_i|$ for any $(y_1,\dots ,y_{n-1},q) \in
\widetilde{\mathbb G_n}$, $1\leq i \leq n-1$, the estimates become sharp for
$\mathbb G_n$ too. We show that these conditions are necessary and sufficient
for $\widetilde{\mathbb G_n}$ when $n=1,2, 3$. In particular for $n=2$, our
results add a few new necessary and sufficient conditions to the existing
Schwarz lemma for the symmetrized bidisc.
