A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains
Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, 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.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \l...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Pal, Sourav [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 |
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| Übergeordnetes Werk: |
Enthalten in: Complex analysis and operator theory - Springer International Publishing, 2007, 16(2022), 5 vom: 11. Juni |
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| Übergeordnetes Werk: |
volume:16 ; year:2022 ; number:5 ; day:11 ; month:06 |
| Links: |
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| DOI / URN: |
10.1007/s11785-022-01242-7 |
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| Katalog-ID: |
OLC2078887889 |
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| 245 | 1 | 0 | |a A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains |
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| 520 | |a Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. | ||
| 650 | 4 | |a Symmetrized polydisc | |
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| 650 | 4 | |a Schwarz lemma | |
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10.1007/s11785-022-01242-7 doi (DE-627)OLC2078887889 (DE-He213)s11785-022-01242-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Pal, Sourav verfasserin aut A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. Symmetrized polydisc Extended symmetrized polydisc Schwarz lemma Roy, Samriddho aut Enthalten in Complex analysis and operator theory Springer International Publishing, 2007 16(2022), 5 vom: 11. Juni (DE-627)565301047 (DE-600)2425163-X (DE-576)409490164 1661-8254 nnns volume:16 year:2022 number:5 day:11 month:06 https://doi.org/10.1007/s11785-022-01242-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 16 2022 5 11 06 |
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10.1007/s11785-022-01242-7 doi (DE-627)OLC2078887889 (DE-He213)s11785-022-01242-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Pal, Sourav verfasserin aut A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. Symmetrized polydisc Extended symmetrized polydisc Schwarz lemma Roy, Samriddho aut Enthalten in Complex analysis and operator theory Springer International Publishing, 2007 16(2022), 5 vom: 11. Juni (DE-627)565301047 (DE-600)2425163-X (DE-576)409490164 1661-8254 nnns volume:16 year:2022 number:5 day:11 month:06 https://doi.org/10.1007/s11785-022-01242-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 16 2022 5 11 06 |
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10.1007/s11785-022-01242-7 doi (DE-627)OLC2078887889 (DE-He213)s11785-022-01242-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Pal, Sourav verfasserin aut A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. Symmetrized polydisc Extended symmetrized polydisc Schwarz lemma Roy, Samriddho aut Enthalten in Complex analysis and operator theory Springer International Publishing, 2007 16(2022), 5 vom: 11. Juni (DE-627)565301047 (DE-600)2425163-X (DE-576)409490164 1661-8254 nnns volume:16 year:2022 number:5 day:11 month:06 https://doi.org/10.1007/s11785-022-01242-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 16 2022 5 11 06 |
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10.1007/s11785-022-01242-7 doi (DE-627)OLC2078887889 (DE-He213)s11785-022-01242-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Pal, Sourav verfasserin aut A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. Symmetrized polydisc Extended symmetrized polydisc Schwarz lemma Roy, Samriddho aut Enthalten in Complex analysis and operator theory Springer International Publishing, 2007 16(2022), 5 vom: 11. Juni (DE-627)565301047 (DE-600)2425163-X (DE-576)409490164 1661-8254 nnns volume:16 year:2022 number:5 day:11 month:06 https://doi.org/10.1007/s11785-022-01242-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 16 2022 5 11 06 |
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10.1007/s11785-022-01242-7 doi (DE-627)OLC2078887889 (DE-He213)s11785-022-01242-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Pal, Sourav verfasserin aut A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. Symmetrized polydisc Extended symmetrized polydisc Schwarz lemma Roy, Samriddho aut Enthalten in Complex analysis and operator theory Springer International Publishing, 2007 16(2022), 5 vom: 11. Juni (DE-627)565301047 (DE-600)2425163-X (DE-576)409490164 1661-8254 nnns volume:16 year:2022 number:5 day:11 month:06 https://doi.org/10.1007/s11785-022-01242-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 16 2022 5 11 06 |
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Enthalten in Complex analysis and operator theory 16(2022), 5 vom: 11. Juni volume:16 year:2022 number:5 day:11 month:06 |
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It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$. 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a schwarz lemma for the symmetrized polydisc via estimates on another family of domains |
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A Schwarz Lemma for the Symmetrized Polydisc Via Estimates on Another Family of Domains |
| abstract |
Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 |
| abstractGer |
Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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$$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 |
| abstract_unstemmed |
Abstract We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc$${\mathbb {G}}_n$$, a family of domains naturally associated with the spectral interpolation, defined by Gn:=∑1≤i≤nzi,∑1≤i<j≤nzizj…,∏i=1nzi:|zi|<1,i=1,…,n.$$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum _{1\le 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}$$We first make a few estimates for the the extended symmetrized polydisc$$\widetilde{{\mathbb {G}}}_n$$, a family of domains introduced in [35] and defined in the following way: G~n:={(y1,…,yn-1,q)∈Cn:q∈D,yj=βj+β¯n-jq,βj∈Cand|βj|+|βn-j|<njforj=1,…,n-1}.$$\begin{aligned} \widetilde{{\mathbb {G}}}_n&:= \Bigg \{ (y_1,\ldots ,y_{n-1}, q)\in {\mathbb {C}}^n :\; q \in {\mathbb {D}}, \; y_j = \beta _j + {\bar{\beta }}_{n-j} q, \; \beta _j \in {\mathbb {C}} \text { and }\\&\quad |\beta _j|+ |\beta _{n-j}| < {n \atopwithdelims ()j} \text { for } j=1,\ldots , n-1 \Bigg \}. \end{aligned}$$We then show that these estimates are sharp and provide a Schwarz lemma for $$\widetilde{{\mathbb {G}}}_n$$. It is easy to verify that $${\mathbb {G}}_n=\widetilde{{\mathbb {G}}}_n$$ for $$n=1,2$$ and that $${{\mathbb {G}}}_n \subsetneq \widetilde{{\mathbb {G}}}_n$$ for $$n\ge 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,\ldots , s_{n-1},p)\in {\mathbb {G}}_n$$, $${n \atopwithdelims ()i}$$ is the least upper bound for $$|s_i|$$, which is same for $$|y_i|$$ for any $$(y_1,\ldots ,y_{n-1},q) \in \widetilde{{\mathbb {G}}_n}$$, $$1\le i \le 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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 |
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