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A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains
Journal
Complex Analysis and Operator Theory
ISSN
16618254
Date Issued
2022-07-01
Author(s)
Pal, Sourav
Roy, Samriddho
Abstract
We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydiscGn, 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 Gn= G~ n for n= 1 , 2 and that Gn⊊ G~ n for n≥ 3. As a consequence of the estimates for Gn~ , we have analogous estimates for Gn. Since for a point (s1, … , sn-1, p) ∈ Gn, (ni) is the least upper bound for | si| , which is same for | yi| for any (y1, … , yn-1, q) ∈ Gn~ , 1 ≤ i≤ n- 1 , the estimates become sharp for Gn too. We show that these conditions are necessary and sufficient for Gn~ 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.
Volume
16
Publication link
Subjects