Assume $f(z)$ is a holomorphic function on the open unit disc $\mathbb{D}$, with $d=sup_{z,w \in \mathbb{D}}|f(z)-f(w)|$, $2|f'(0)|=d$, and $f(0)=0$.
What I'm trying to do is to show: $sup_{z \in \mathbb{D}}|f(z)|\le\frac{d}{2}$
The motivation of this question is from the exercise 7, chapter 2, on Stein's Complex analysis. It said that such a function with $f(0)$ may not be $0$ is linear (but I haven't work it out yet). I think since that statement is true, then my question is just a nature clorollary, but it seems not easy to figure out.
