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Let $f$ be an analytic function on $|z|\leq 1$ with $f(0)=0$ and let $|f(z)|$ have a maximum for $|z|\leq 1$ at $z_0=1$. Show that $f^{'}(z_0)\neq 0$ unless $f$ is constant.

Here is what I have: If $|f(1)|=0$ then $f$ is constant and we are done. So assume $|f(1)|>0$. Then $f(1)\neq 0$ and the function $g(z)=\frac{f(z)}{f(1)}$ is analytic. Also $g$ satisfies the conditions of Schwarz Lemma.

Here is where I am stuck. I want to get $g(z)=cz$ for some $c$ with $|c|=1$.

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    Theorem 4.1 in this link may be useful http://arxiv.org/pdf/1001.1805v1.pdf2012-07-29

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