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I wish to prove that all Möbius transformation raking the unit disk into itself are of the form $k\frac{z-l}{1-z\bar{l}}$ where $|k| = 1$.

More specifically, I ask, in addition to the main question above, the maximum modulus principle implies that the analytic continuation of such mobius transform has to carry the unit circle to itself.

I realize that numerous posts like this and this already seems to have an answer here, but I couldn't find a satisfactory answer yet. Letting $T(z)$ be an arbitrary Möbius transformation that preserves the unit disk, and writing $T(z) = \frac{az+b}{cz+d}$ with (without loss of generality) $a=0$ or $a=1$, we must somehow show that $T$ is of the form specified above... It seems like we have to use the maximum modulus principle, yet all the answers from other posts did not elaborate on how this would be used.

Please note that my question only asks for one direction and not the other, and does NOT ask to show that all conformal automorphism of the unit disk are of this form.

Squirrel-Power
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  • @JoséCarlosSantos Thanks, but I already looked at that post and couldn't understand neither of HagenvonEitzen's or WillJagy's answer, which are (and you can check this) up to Oct 28, 2018 the only 2 answers on the post that specifically answers my question (notice also that the question from the post you mentioned was changed midway so some people's answers aren't applicable no more). – Squirrel-Power Oct 28 '18 at 21:49
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    That doesn't change the fact that this is a duplicate question. – José Carlos Santos Oct 28 '18 at 21:51
  • You've clearly thought about the problem, and I like your idea about possible application of the maximum modulus principle. But it is a duplicate of the earlier Question as currently stated. Perhaps you can identify a step in one of the Answers to that which gives you difficulty, and make that step the focus of this Question? – hardmath Oct 28 '18 at 21:57

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