Suppose that f is holomorphic and satisfies $|f(z)|≤1$for all $|z|<1$. Then I want to prove that if $f$ has zero of order $m$ at $z_0$, then $|z_0| ≥ |f(0)|^{(1/m)}.$
I guess it can be done by Schwarz's lemma.
Any help will be appreciated.
Thank you so much.
Let $$ F(z)=\frac{f(z)}{\Big(\displaystyle\frac{z-z_0}{1-\bar{z_0}z}\Big)^m}.$$ Then $F(z)$ is holomorphic and satisfies $|F(z)|\le 1$ for all $|z|<1$. Therefore we have $|F(0)|\le 1$ which yields $$ \frac{|f(0)|}{|z_0|^m}\le 1$$ and hence we have $$ |z_0|\ge |f(0)|^{(1/m)}.$$
