This question is exercise 8.2 in Conway's Functions of One Complex Variable I. It states:
Let $\mathbb{D}\subset\mathbb{C}$ be the open unit disk, and let $K=\{z\in\mathbb{D}: \frac{1}{4}\leq |z|\leq \frac{3}{4}\}\subset\mathbb{D}$ compact. Find a holomorphic function $f$ on $K$ (so on a neighborhood of $K$), which cannot be approximated on $K$ by holomorphic functions on $\mathbb{D}$.
After this, Conway makes the remark:
The next two problems are concerned with the following question. Given a compact set $K$ contained in $G$ open, can holomorphic functions on $K$ be approximated on $K$ (with the $\sup$ norm) by holomorphic functions on $G$? Exercise 2 says that for an arbitrary choice of $K$ and $G$, this is not true. Exercise 4 below gives criteria for a fixed $K$ and $G$ such that this can be done. Exercise 3 is a lemma which is useful in proving Exercise 4.
Exercise $3$ is a lemma, and Exercise 4 is the statement of Runge's theorem that was covered in my complex analysis course:
Let $G \subset \mathbb{C}$ be an open set and let $K\subset G$ be a compact subset. Then the following conditions are equivalent.
- Every holomorphic function on a neighborhood of $K$ can be approximated uniformly on $K$ by holomorphic functions on $G$.
- The open set $G\setminus K$ has no component whose closure in $G$ is compact.
- For every $z\in G\setminus K$ there exists a holomorphic function on $G$ such that: $$\sup_K |f| < |f(z)|$$
So, is there any suggestions for this problem? I couldn't think of exactly how to construct such a function. My guess would be something standard like $f(z)=1/z$, but I'm not sure how to show that this cannot be approximated on $K$.