A problem about a sequence of polynomials Let $\Omega = \{z: |z|<1 $and $|2z-1|>1\}$, and suppose $f \in H(\Omega)$. 
a) Must there exist a sequence of polynomials $P_n$ such that $P_n \rightarrow f$ uniformly on compact subsets of $\Omega$.
b) Must there exists such a sequence which converges to f uniformly in $\Omega$.
c) Is the answer to b) changed if we require more of f , namely , that f be holomorphic in some open set which contains the closure of $\Omega$.
Which theorem or result will help me to solve this one. Any hints are most welcomed.
 A: $\Omega$ is the region between the two circles $\lvert z\rvert = 1$ and $\left\lvert z - \frac{1}{2}\right\rvert = \frac{1}{2}$. These two circles touch at $z = 1$ and have a common tangent there.
For $n > 2$, let $K_n = \left\{ z\in \Omega : \operatorname{dist}(z,\partial\Omega) \geqslant \frac{1}{n}\right\}$ Then $K_n$ is a compact subset of $\Omega$, and every compact $K\subset \Omega$ is contained in $K_n$ for all large enough $n$. Since $\mathbb{C}\setminus K_n$ is connected, Runge's theorem asserts the existence of a polynomial $P_n$ such that
$$\max_{z\in K_n} \lvert P_n(z) - f(z)\rvert < \frac{1}{n}.$$
Thus $P_n \to f$ uniformly on all compact subsets of $\Omega$, hence the answer to question a) is yes.
Since polynomials are continuous on all of $\mathbb{C}$, we have
$$\sup_{z\in\Omega} \lvert P(z) - Q(z)\rvert = \max_{z \in \overline{\Omega}} \lvert P(z) - Q(z)\rvert$$
for all polynomials $P,Q$, and hence if a sequence of polynomials converges uniformly on $\Omega$, it converges uniformly on $\overline{\Omega}$. In particular, the sequence converges uniformly on the unit circle $\lvert z\rvert = 1$. But by the maximum modulus principle, the sequence then converges uniformly on the closed unit disk $\lvert z\rvert \leqslant 1$. Thus a necessary condition for the existence of a sequence of polynomials converging to $f$ uniformly on $\Omega$ is that $f$ is the restriction of a function that is holomorphic on the whole unit disk, and continuous on the closed unit disk. Not all $f\in H(\Omega)$ satisfy that, e.g. there is no sequence of polynomials converging uniformly on $\Omega$ to
$$f(z) = \frac{1}{z-\frac{1}{2}}.$$
Thus the answer to question b) is no.
Since the example above is holomorphic in a neighbourhood of $\overline{\Omega}$, the answer to c) is also no.
