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Each analytic function mapping the right half complex plane into itself must satisfy $$ \left|\frac{f(z)-f(1)}{f(z)+f(1)}\right| \leqslant \left|\frac {z-1}{z+1} \right|$$ for $\text{Re}\; z > 0.$

I have a hunch that this is an application of Schwarz's Lemma. I don't know how to proceed though. Thanks in advance.

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The map $h$ from $\{z,\Re z>0\}$ to the open unit disk $D$ given by $h(z)=\frac{z-1}{z+1}$ is one-to-one, then use Schwarz lemma with $g(z):=\dfrac{f\left(\frac{1+z}{1-z}\right)-f(1)}{f\left(\frac{1+z}{1-z}\right)+f(1)}$.

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this is actually a past qual question. I am thinking $f$o$h^{-1}$ is actually a mapping from unit disk to right half plane, but don't we need the mapping from unit disk to unit disk in Swartz's Lemma? I am confused. Could you please elaborate? –  Deepak Dec 29 '12 at 1:11
    
,But, I still have difficulty seeing ${h\circ}f\circ h^{-1}(0)=0$. I can see after that Swartz lemma gives the result. –  Deepak Dec 30 '12 at 3:43
    
But, in what ground can we assume that? @Davide Giraudo –  Deepak Dec 30 '12 at 17:23
    
I don't know if you can access this because this is the departmental website, you can find it here if the link works sierra.nmsu.edu/dept/Comp%20Exams/Complex%20Analysis/2001.pdf –  Deepak Dec 30 '12 at 19:13

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