# Does bounded holomorphic function on the unit disc with bounded diameter is bounded by its radius?

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.

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