Suppose $f:\mathbb{D}\to \mathbb{C}$ is holomorphic and the diameter d:= $sup_{z,w\in \mathbb {D}} \vert{f(z)-f(w)}\vert $ of the image of $f$. We know that 2$\vert f'(0) \vert = d $ if and only if $f$ is linear.
If $f(z)=z+\textbf{a}z^2$ for some nonzero $\textbf{a}\in \mathbb{C}$, then $f$ is not linear. We know that $f'(0)=1$. Then, the diameter of $f$ >2 because 2$\vert f'(0) \vert \leq d$ always satisfy. However, if $\vert \textbf{a} \vert$ be very small, I couldn't find the points $z,w \in \mathbb{\bar{D}}$ satisfying d(the diameter of $f$) > 2.
Any help is appreciated..!
Thank you.