A question about Maximum Principle in complex analysis Let $f$ be holomorphic on open set containing $\overline{D}$,prove that there exists $z_{0}\in \partial D$, such that  $|\dfrac{1}{z_{0}}-f\left(z_{0}\right)|\geq 1$
This problem can be solved by maximum principle,but I don't know how to select a proper function to apply this theorem.

Define  $g(z) = 1 - z \, f(z)$ in the unit disk
and note that $g(0) = 1$. What does the maximum principle imply?
