SoatDev IT Consulting
SoatDev IT Consulting
  • About us
  • Expertise
  • Services
  • How it works
  • Contact Us
  • News
  • November 18, 2023
  • Rss Fetcher

Let D be the open unit disk in the complex plane. The Schwarz lemma says that if f is an analytic function from D to D with f(0) = 0, then

|f(z)| leq |z|

for all z in D. (The lemma also says more, but this post will focus on just this portion of the theorem.

The Schwarz-Pick theorem generalizes the Schwarz lemma by not requiring the origin to be fixed. That is, it says that if f is an analytic function from D to D then

left| frac{f(z) - f(w)}{1 - f(z),overline{f(w)}} right| leq left| frac{z - w}{1 - z,overline{w}}right|

The Schwarz-Pick theorem also concludes more, but again we’re focusing on part of the theorem here. Note that if f(0) = 0 then the Schwarz-Pick theorem reduces to the Schwarz lemma.

The Schwarz lemma is a sort of contraction theorem. Assuming f(0) = 0, the lemma says

|f(z) - f(0)| leq |z - 0|

This says applying f to a point cannot move the point further from 0. That’s interesting, but it would be more interesting if we could say f is a contraction in general, not just with respect to 0. That is indeed what the Schwarz-Pick theorem does, though with respect to a new metric.

For any two points z and w in the open unit disk D, define the Poincaré distance between z and w by

d(z,w) = tanh^{-1}left( left| frac{z - w}{1 - zoverline{w}}right| right)

It’s not obvious that this is a metric, but it really is. As is often the case, most of the properties of a metric are simple to confirm, but the proving the triangle inequality is the hard part.

If we apply the monotone function tanh-1 to both sides of the Schwarz-Pick theorem, then we have that any analytic function f from D to D is a contraction on D with respect to the Poincaré metric.

Here we’re using “contraction” in the lose sense. It would be more explicit to say that f is a non-expansive map. Applying f to a pair of points may not bring the points closer together, but it cannot move them any further apart (with respect to the Poincaré metric).

By using the Poincaré metric, we turn the unit disk into a hyperbolic space. That is D with the metric d is a model of the hyperbolic plane.

Related posts

  • Kepler and the contraction mapping theorem
  • Fixed points
  • Hyperbolic tangent sum

The post Schwarz lemma, Schwarz-Pick theorem, and Poincare metric first appeared on John D. Cook.

Previous Post
Next Post

Recent Posts

  • Cloudflare launches a marketplace that lets websites charge AI bots for scraping
  • Tinder’s mandatory facial recognition check comes to the US
  • 5 Must-Know Tips to Secure Your Smart Home Devices
  • Leading South Africa into the Age of AI: What Executives Must Do Now!
  • How Can AI Cultivate the Future of Agriculture

Categories

  • Industry News
  • Programming
  • RSS Fetched Articles
  • Uncategorized

Archives

  • July 2025
  • June 2025
  • May 2025
  • April 2025
  • February 2025
  • January 2025
  • December 2024
  • November 2024
  • October 2024
  • September 2024
  • August 2024
  • July 2024
  • June 2024
  • May 2024
  • April 2024
  • March 2024
  • February 2024
  • January 2024
  • December 2023
  • November 2023
  • October 2023
  • September 2023
  • August 2023
  • July 2023
  • June 2023
  • May 2023
  • April 2023

Tap into the power of Microservices, MVC Architecture, Cloud, Containers, UML, and Scrum methodologies to bolster your project planning, execution, and application development processes.

Solutions

  • IT Consultation
  • Agile Transformation
  • Software Development
  • DevOps & CI/CD

Regions Covered

  • Montreal
  • New York
  • Paris
  • Mauritius
  • Abidjan
  • Dakar

Subscribe to Newsletter

Join our monthly newsletter subscribers to get the latest news and insights.

© Copyright 2023. All Rights Reserved by Soatdev IT Consulting Inc.