I know the following is a well-known result.


Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup\_{z, w \in D} |f(z)-f(w)|$$ 
Furthermore, there is equality if and only if $f$ is linear.


I need some reference about the second part, i.e. there is equality if and only if $f$ is linear.



$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear
This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the *paper Area, capacity and diameter versions of Schwarz's lemma* by Burckel, Marshall, Minda, Poggi-Corradini and Ransford.


See Theorem 1.3 [here](http://arxiv.org/pdf/0801.3629v1.pdf)


