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 G(n), a family of domains naturally associated with the spectral interpolation, defined by G(n) := {(Sigma(1 <= i <= n) z(i) Sigma(1 <= i<j <= n) z(i)z(j) ..., Pi(n)(i=1)z(i)) : vertical bar z(i)vertical bar < 1, i = 1, ..., n}. We first make a few estimates for the the extended symmetrized polydisc <(G)over tilde>(n) , a family of domains introduced in [35] and defined in the following way: (G) over tilde (n) := {(y(1), ..., y(n-1),q) is an element of C-n : q is an element of D, y(j) = beta(j) + (beta) over bar (n-j)q, beta(j) is an element of C and vertical bar beta(j)vertical bar + vertical bar beta(n-j)vertical bar < ((n)(j)) for j = 1, ..., n - 1}. We then show that these estimates are sharp and provide a Schwarz lemma for (G) over tilde (n). It is easy to verify that G(n) = (G) over tilde (n )for n = 1, 2 and that G(n) not subset of (G) over tilde (n) for n >= 3. As a consequence of the estimates for (G) over tilde (n) we have analogous estimates for G(n). Since for a point(s(1), ..., s(n-1), p) is an element of G(n),((n)(i)) is the least upper bound for vertical bar s(i)vertical bar, which is same for vertical bar y(i)vertical bar for any (y(1), ..., y(n-1), q) is an element of (G) over tilden 1 <= i <= n-1, the estimates become sharp for G(n) too. We show that these conditions are necessary and sufficient for (G) over tilden 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.