Can bounded analytic function defined on a simply connected region be approximated by bounded polynomials? Let $G$ be a simply connected bounded region in the complex plane (say the unit disk centered at $0$). Suppose $f$ is analytic on $G$, then $f$ can be locally uniformly approximated by polynomials (just take truncations of its power series).
Question: is it true that if  $f$ is bounded on $G$, it can be approximated by polynomials $\{p_n\}$ on the whole region $G$ such that $\sup_n\sup_{z\in G}|p_n(z)|＜\infty$?
By Runge's theorem, global polynomial approximation exists on simply connected domains. But I am not sure how to determine boundedness..
 A: This is true for the circle (one needs Caratheodory's kernel theorem to prove it), but not in general.
To see why, let $A:=\mathbb{D}-[0,1)$ open and  simply connected (since its complement is connected). There exists a conformal map $\varphi: \mathbb{D}\to A$ and it admits a continuous extension to $\overline{\mathbb{D}}\to \overline{A}$ by Carathéodory-Torhorst. Note in particular that $\varphi^{-1}$ is bounded.
Define now $f$ as a Blaschke product (thus bounded) having $X=\cup\{(1-1/n^2)e^{iq_n}\}$ as zeros (where $q_n$ is an enumeration of the rationals in $[0,2\pi)$).
This function is analytic bounded on $\mathbb{D}$ and non-extendible.
Since $\varphi$ has a continuous extension to $\overline A$, $\overline{\varphi(X)}
\supseteq\varphi(\overline{X})\supset \varphi(\partial\mathbb{D})=\partial A$.
Now, let $g=f\circ \varphi^{-1}$, which is clearly bounded and holomorphic on $A$. Suppose thre exists a sequence of bounded polynomials $p_n\to f\circ\varphi$. By continuity, $\{p_n\}$ must be bounded on $\overline{A}=\overline{\mathbb{D}}$. Montel's theorem implies the existence of a converging subsequence $p_{n_k}$. The limit must be an extension of $f\circ\varphi^{-1}$, and it must be analytic on $\mathbb{D}$.
Since $[0,1)\subset \overline{g^{-1}(0)}$, the identity principle implies $g\equiv 0$, a contradiction.
This is true in general: a bounded function on an open connected set $A$ is the limit of a bounded sequence of polynomials only if it admits a bounded holomorphic extension to $\hat{\mathbb{C}}-
\overline{(\hat{\mathbb{C}}-\overline{A})}$. This is actually a iff but the other direction of the implication is longer.
