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.