Analytic function property

Question

How to prove the following:

Let f be analytic on $\mathbb{D}$ and $a\in\mathbb{D}$. Then $f'(a)=0$ iff $$\left|\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right|\leq \left|\frac{a-z}{1-\overline{a}z}\right|^2.$$

Any hint is welcome. Thanks in advance.

Answer 1

Here is an idea which doesn't solve your problem, but may help you to prove what you need. Define $$\varphi_w(z)= \frac{w-z}{1-z\bar{w}},$$ which is abijection from the unit disk $\mathbb{D}$ to itself. Also $\varphi_w (\varphi_w(z))=z$, $\varphi_w (w)=0$, and $\varphi^{-1}_w (0)=w$. Conside the function $$g(z)=\varphi_w o f o \varphi^{-1}(z).$$ Then $g$ is analytic and $g(0)=0$. By Schwartz's lemma $$|g(z')|\leq |\varphi_w o f o \varphi^{-1}(z')|\leq |z'|.$$ For $z\in \mathbb{D}$ let $z'=\varphi_w^{-1}(z)$. Then $$|\varphi_w o f (z)|\leq |\varphi_w^{-1}(z)|.$$ The inequality you are looking for, without the square in the right hand side, follows immediately from the above inequality with $w=f(a)$. You probably need a modified/improves Schwartz's inequality using the assumption $f'(a)=0$ to prove what you want.