Vipul: 2 revisions

2008-05-18T19:18:21Z

2 revisions

← Older revision	Revision as of 19:18, 18 May 2008
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Vipul: Schwarz-Pick lemma for disc moved to Schwarz-Pick lemma for disk

2008-04-26T18:55:44Z

Schwarz-Pick lemma for disc moved to Schwarz-Pick lemma for disk

← Older revision	Revision as of 18:55, 26 April 2008
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Vipul: New page: ==Statement== Let $D$ denote the open unit disc in $\mathbb{C}$ . Consider the function $\Delta:D \times D \to \R$ , given by: $\Delta(z,w) := \fr...$

2008-04-20T23:09:28Z

New page: ==Statement== Let <math>D</math> denote the open unit disc in <math>\mathbb{C}</math>. Consider the function <math>\Delta:D \times D \to \R</math>, given by: <math>\Delta(z,w) := \fr...

New page

==Statement==

Let <math>D</math> denote the [[open unit disc]] in <math>\mathbb{C}</math>. Consider the function <math>\Delta:D \times D \to \R</math>, given by:

<math>\Delta(z,w) := \frac{|z - w|}{|\overline{w}z - 1|}</math>

Then, if <math>f: D \to D</math> is a holomorphic function, we have:

* For all <math>z,w \in D</math>, <math>\Delta(f(z),f(w)) \le \Delta(z,w)</math>
* <math>f</math> is a conformal automorphism of <math>D</math> if and only if for all <math>z,w \in D</math>, we have <math>\Delta(f(z),f(w)) = \Delta(z,w)</math>

==Facts used==

* [[Schwarz lemma]]

==Proof==

{{fillin}}

Schwarz-Pick lemma for disk - Revision history

Vipul: 2 revisions

Vipul: Schwarz-Pick lemma for disc moved to Schwarz-Pick lemma for disk

Vipul: New page: ==Statement== Let D denote the open unit disc in \mathbb{C}. Consider the function \Delta:D \times D \to \R, given by: \Delta(z,w) := \fr...

Vipul: New page: ==Statement== Let $D$ denote the open unit disc in $\mathbb{C}$ . Consider the function $\Delta:D \times D \to \R$ , given by: $\Delta(z,w) := \fr...$