This is a homework problem, so my apologies for a seeming lack of motivation. I want to understand what's going on in this problem more than I want someone to just write down the solution, so if someone could spell out for me the underpinnings of what I'm missing, that'd be best.
1) Consider the region defined by $\Omega = \{ z \in \mathbb C: 1< |z| < 2$ or $4 < |z| < 5 \}$. Let $f$ be a holomorphic function on $\Omega$, and $K$ be compactly contained in $\Omega$. Show that there is a function $g$ holomorphic everywhere but $0$ and $3$ so that $\sup_K |f - g | < 1/2$.
This part of the problem I was able to do using Runge's theorem. Staring at the location of the points, and the region in question, we see that it contains a point in each bounded region of the complement of $\Omega$, so we can come up with a sequence of rational functions with poles at $0$ and $3$ which uniformly approximate $f$. Then we just choose a member of this sequence so that the approximation is sufficiently close.
The next part of the problem is what I find challenging.
2)Now redefine $K = \{ z \in \mathbb C: 4/3 \leq |z| \leq 3/2$ or $41/10 \leq |z| \leq 45/10 \}$.
The question is to find a function $f$ which is holomorphic on $\Omega$ so there does NOT exist a holomorphic function $g: \mathbb C - \{ 0 \} \rightarrow \mathbb C$ where $\sup_K |f-g| \leq 1$.
Staring at this sub-double-annulus (wow that's a mouthful), it seems hard to know what to do. My intuition says to assume that such a function $g$ does exist, and then try to find some contradiction. One thought I had was to try to integrate $f-g$ over some closed curve, and use the fact that the sup is bounded to estimate the magnitude of the integral. I tried an approach like this with a few functions $f$, looking for things that made the integral kind of simple. In each case, I would run into a problem because no matter how I tried to compute the integrals, I'd end up not knowing enough about $g$ in order to say that the estimate didn't work...
Can anyone point me in the right direction?
Hint: Let $f(z) = c/(z-3),$ with $c$ to be determined. Let $g$ be holomorphic on $\mathbb C \setminus \{0\}.$ Consider
$$\int_{C_2} (f(z)-g(z))\, dz \,- \int_{C_1} (f(z)-g(z))\, dz $$
for well chosen circles $C_1,C_2$ lying in $K.$ If $|c|$ is large, the answer you get above will be incompatible with $|f-g| \le 1$ on $K.$