Max Mod Principle

Question

I'm stuck with the following exercise:

Let $f$ be holomorphic on an open set containing $\bar{D}$, the closed unit disk. Prove that there exists a $z_0 \in \partial D$ such that

$$\bigg|\frac{1}{z_0} - f(z_0)\bigg| \ge 1 $$

I know that I need to use the maximum modulus principle, but I am not sure where to begin, besides proving by contradiction.

Answer 1

I give you more detail beyond David C. Ullrich's post. In fact, you don't have to use maximum modulus principle explicitly. But you need to use the idea to prove it. We have $$ \int_\gamma \left(\frac1{z} - f(z)\right)\,dz=\int_\gamma \frac1{z} \,dz=2\pi i\tag{1} $$ Use contradiction to prove that if condition were not true, $(1)$ would not be held.

Suppose for all $z \in \partial D$, there is $|1/z - f(z)| < 1$. Then $$\left|\int_\gamma(1/z - f(z))\,dz\right|\leqslant\int_\gamma|1/z - f(z)|\,dz<2\pi$$

Answer 2

Hint: Define $g(z)=1/z - f(z)$. Say $\gamma$ is the unit circle. What can you say about $$\int_\gamma g(z)\,dz\,\,?$$