$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

complex-analysis analysis reference-request derivatives

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear.

I know this is a well-known result. Where can I find the proof ?


See "On the greatest oscillation of an analytic function in a circle": http://manetheren.bigw.org/~ray/diampblm.pdf

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