This question was asked in complex analysis exam of previous year which I got from a senior and I am not able to solve it.
Let $f: U\to U$ be a holomorphic function with $f(0) =0 =f(a)$ where $a \in U\setminus\{0\}$ . Show that $|f'(0) |\leq |a|$.
Here $U$ is the unit disc. Schwartz lemma proves $|f'(0) | \leq 1$ and as $a<1$ , so it is proved easily.
Is this proof true?
If not then kindly tell me right proof.
Best Answer
Let $$g(z):=\frac{f(z)(1-z\overline{a})}{z-a}$$ It's not hard to prove that $g:U\to U$ (hint: use the maximum modulus principle) and it is easy to see that $g(0)=0$. Thus, by Schwarz' lemma,
$$\begin{align*}1\ge &|g'(0)|=\left|\frac{f'(0)}{a}\right|\\ |a|\ge&|f'(0)|\end{align*}$$