Consider the following problem.
If $D=\{z\in\mathbb{C}||z|<1\}$, and $f:D\rightarrow D$ is analytic with $f\left(\frac{2}{3}\right)=\frac{1}{3}$, then find the maximum value of $|f'\left(\frac{2}{3}\right)|$.
I am clue less as to how to proceed with this question.
I have tried thinking in terms of Cauchy's formula and the inequality arising out of it. But, that is given for a function with known bound.
I also went through the book, bak and newman complex analysis which is mentioned in the post Ununderstood estimating of $|f({3\over 4})|$. However, this method also requires the function $f$ to be bounded.
Please give a hint on how to proceed with this.