Schwarz reflection principle and linear fractional transformations

[arthurhenry]

The following comes from the complex analysis text by Joseph Bak:



He is trying to determine all conformal mappings f of upper half plane H onto the unit disk.

"Let us first assume that f is an LFT and f(a)=0 for Im(a)>0.
Then, since the real axis is mapped into the unit circle, it follows by Schwarz Principle of reflection that f(a*)=Infinity, so that...

Here a* denotes the conjugate of a.

I am having difficulty following how the Schwarz Reflection Principle is applied...and we get that f(a*)=Infinity.

thank you.

Also, Isn't true that the inverse Caylet Transform map the unit circle into real axis.If so, why "there is no non-constant analytic function in the unit disk that is real valued on the unit circle" ?
Thank you

 

[fresh_42]

Here are some links which might help, since I haven't Bak's book.

Schwarz’s reflection principle
http://www.math.ust.hk/~machiang/5030/notes/5030_notes_2017_18_30Oct2017_18.pdf (p.123f)

Cayley transform
https://en.wikipedia.org/wiki/Cayley_transform
https://lcvmwww.epfl.ch/teaching/modelling_dna/index.php?dir=exercises&file=serie03.pdf (properties)