Let $f:\mathbb{D}\to \mathbb{C}$ be a analytic with $|f(z)|<1$ .Prove that for all $a,b \in \mathbb{D}$ that $$\left|\frac{f(a)-f(b)}{1-\overline{f(b)}f(a)}\right|\le\left |\frac{a-b}{1-\bar{b}a}\right|.$$
What I tried
I am trying to prove by Schwarz lemma for the function $g(z)=\phi_{f(a)}(f(\phi_{-a}(z)))$ where $\phi_a(z)=\frac{z-a}{1-\bar{a}z}$, but I didn't get the result. Can any one help? Thank you.
Hint. We have that $g:\mathbb{D}\to \mathbb{D}$ and $g(0)=0$, therefore by Schwarz Lemma, for all $z\in \mathbb{D}$, $$|g(z)|=|\phi_{f(a)}(f(\phi_{-a}(z)))|\leq |z|.$$ Now note that $\phi_{a}:\mathbb{D}\to \mathbb{D}$ is a bijection and $\phi_{-a}^{-1}=\phi_{a}$. Then let $z=\phi_{-a}^{-1}(b)=\phi_{a}(b)$.