Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Automorphisms of the unit disc is less than 1

  1. Jun 5, 2014 #1
    I want to show that the modulus of the automorphism

    [tex]\frac{a-z}{1-\overline{a}z}[/tex]

    is strictly bounded by 1 in the unit disc. Applying Schwarz lemma gives the result immediately. But I am looking for a straight forward proof for that.

    Thanks in advance
     
    Last edited: Jun 5, 2014
  2. jcsd
  3. Jun 5, 2014 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    So you need to prove that ##|a-z| \leq |1 - \overline{a}z|## whenever ##|z| \leq 1##. Equivalently, you require
    $$(a-z)(\overline{a}-\overline{z}) \leq (1 - \overline{a}z)(1 - a\overline{z})$$
    Performing the multiplication on both sides, we need
    $$|a|^2 - 2\text{Re}(\overline{a}z) + |z|^2 \leq 1 - 2\text{Re}(\overline{a}z) + |a|^2|z|^2$$
    It should be straightforward from this point.
     
  4. Jun 5, 2014 #3
    Now it is clear, and the last one is true because
    [tex](1-|z|^2)(1-|a|^2)>0.[/tex]
    Thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Automorphisms of the unit disc is less than 1
Loading...