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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.

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  • $\begingroup$ Be aware $d$ may be $\infty$. Technical detail to keep in mind. $\endgroup$ – Brevan Ellefsen May 4 '19 at 5:51
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    $\begingroup$ Your actual question is answer here: math.stackexchange.com/q/339569/42969 $\endgroup$ – Martin R May 4 '19 at 10:11
  • $\begingroup$ @MartinR Thanks! $\endgroup$ – C.X.Neo May 6 '19 at 2:18

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