# Thread: Cpx Analysis: Lower bound on a derivative.

1. ## Cpx Analysis: Lower bound on a derivative.

The Problem:

Suppose that $\Omega$ is a simply connected domain with $\Omega \ne \mathbb{C}$, $a \in \Omega$, and for some $r > 0, D(a;r) \subseteq \Omega$ (an open disk at a with radius r). For $\mathbb{D}$ a unit disk, let $f: \mathbb{D} \rightarrow \Omega$ be a conformal map with $f(0) = a$. Prove that $|f'(0)| \ge r$.

Ideas:

- There is a drawing (.pdf) attached that represents the set context.

- Application of the Riemann Ext Thm. provides a conformal map $h1$ from $\Omega$ onto $\mathbb{D}$ with $h1(a) = 0$ and $h1'(a) > 0$ (or another suitable rotation) and a map $h2$ from $f(\mathbb{D})$ onto $\mathbb{D}, h2(a) = 0$ and $h2'(a) > 0$.

- Seems like a setup for Schwarz's Lemma. For example, we can take $g = h1 \circ f$ with $g(0) = 0$ and $g(\mathbb{D}) \subseteq \mathbb{D}$. Then the lemma asserts that $|g(z)| \le |z|$ and $|g'(0)| \le 1$. So $|h1'(f(0))| |f'(0)| \le 1 \Rightarrow |h1'(a)| |f'(0)| \le 1$. I don't see where this helps with this proof.

Have you considered the map $g(z)=\frac{1}{r}(z-a)$, which takes $D(a,r)$ to $\mathbb{D}$?