Does there exist an analytic function $f$ from unit disc $D$ to itself such that $f(0)= \frac{1}{2}$ and $f'(0)= \frac{2}{3}$ ?
I know such function exists , since $ |f'(a)| \leq \frac {1- |f(a)|^2}{1 - |a|^2} $ is satisfied for $ a = 0$ in $D$, but how to find such function?
Choose $\beta$ with $0<|\beta|\leq 1$ and $a$ with $|a|<1$ and consider $f:D\to D$ defined by $$f(z)=\beta\frac{z-a}{1-\overline az},|z|\leq 1.$$Try to find $a$ and $\beta$ from the values of $f(0)$ and $f'(0)$.
After some calculations $\beta=\frac{\sqrt{13}+2}{6}$ and $a=-\frac{3}{\sqrt{13}+2}$ work.
