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May you help me with this question:

Let $F$ a familiy of analytic functions at $H=\{z:\mathrm{Im}z>0\}$ that satisfies $f(i)=0$ and $|f(z)|\leq 1$.

I want to find $\sup\limits_{f \in F}|f(2i)|$.

I think that the solution is based on Möbius transformation.

Thank you.

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