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 polydiscG_n, a family of domains naturally associated with the spectral interpolation, defined by Gn:={(∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi):|zi|<1,i=1,…,n}.We first make a few estimates for the the extended symmetrized polydiscG~ _n, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<(nj)forj=1,…,n-1}.We then show that these estimates are sharp and provide a Schwarz lemma for G~ _n. It is easy to verify that G_n= G~ _n for n= 1 , 2 and that G_n⊊ G~ _n for n≥ 3. As a consequence of the estimates for G_n~ , we have analogous estimates for G_n. Since for a point (s₁, … , s_n-1, p) ∈ G_n, (ni) is the least upper bound for | s_i| , which is same for | y_i| for any (y₁, … , y_n-1, q) ∈ G_n~ , 1 ≤ i≤ n- 1 , the estimates become sharp for G_n too. We show that these conditions are necessary and sufficient for 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.