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Automorphisms of the unit disc is less than 1

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


    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


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    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
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