Let $\mathbb{D}$ denote the unit disk in the complex plane $\mathbb{C}$. Suppose that $f: \mathbb{D}\rightarrow\mathbb{D}$ is holomorphic and $f(0)=0$, $f'(0)=a>0$. How can I prove that $f$ is biholomorphic in $B(0,\frac{a}{1+\sqrt{1-a^2}})$? Moreover, what is the maximal radius $R$ such that $f$ is biholomorphic in $B(0,R)$?
Here is where I go. Let $M(R)=\sup_{|z|=R}|f(z)|$. Consider the holomorphic function $g(z)=\frac{f(Rz)}{M(R)}:\mathbb{D}\rightarrow \mathbb{D}$. Then $g(0)=0$ and $g'(0)=\frac{Ra}{M(R)}$. Hence Schwartz's lemma gives $Ra\leq M(R)$. Sadly, the only easimate I can get about $M(R)$ is that $M(R)\leq 1$. Since $a\leq 1$, this can not lead to an inverse inequality when $R=\frac{a}{1+\sqrt{1-a^2}}$.
I think I may need some other techniques. Any advice will be helpful.