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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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    Great question! I'm asking a similar question here, which might be a more general result: http://math.stackexchange.com/questions/1245940/prove-that-big-fracfz-fwfz-overlinefw-big-le-big-fracz-w2015-04-22

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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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    I'm asking a similar question here: http://math.stackexchange.com/questions/1245940/prove-that-big-fracfz-fwfz-overlinefw-big-le-big-fracz-w2015-04-22