Let $K\subset \mathbb{C}$ be a compact subset. The question here is to show that $\mathbb{C}\backslash K$ is connected if and only if for every $w\in \mathbb{C}\backslash K$ there exists polynomial $f(z)$ such that $|f(w)|>\sup_{w\in K} |f(z)|$.
I've attempted the above problem with very little success rate. Let us suppose $\mathbb{C}\backslash K$ is indeed connected. So for any $w\in \mathbb{C}\backslash K$ we may choose a neighbourhood $U_w\subset \mathbb{C}\backslash K$ containing $w$. (We can do so since $\mathbb{C}\backslash K$ is open) Then applying Runge's theorem we get that for any $f$ holomorphic on open subset containing $K$ we have some rational function with poles only in $U_W$ satisfying $\sup_{z\in K} |f(z)-R(z)| <\varepsilon$. I believe we need to make some clever choice of $f$ for this result to hold. Any hints (as well as for the converse claim) would be greatly appreciated.
Best Answer
Hints: First the converse, which is more elementary. In general $\Bbb C\setminus K$ has exactly one bounded component, so if it is not connected it has a bounded component $C$. If you show that $\partial C\subset K$ then the existence of a polynomial as above for $w\in C$ would contradict the maxium modulus theorem.
Now suppose $\Bbb C\setminus K$ is connected and $w\in\Bbb C\setminus K$. Let $K'=K\cup\{w\}$. Show that $\Bbb C\setminus K'$ is connected. Hence Runge's theorem shows that any function $g$ holomorphic in a neighborhood of $K'$ can be uniformly approximated on $K'$ by a polynomial $f$; the right choice of $g$ and $\epsilon$ gives the $f$ you want.