Lemma 11.16.100BCE16BE996

Proof. Write $h$ for the function and $\Delta _ K,\Delta _ L$ for the disks. Their union is polynomially convex, so testing with polynomials in $h$ shows that $h$ maps the holomorphic hull of $K\cup L$ into $\Delta _ K\cup \Delta _ L$. Suppose that $p\notin K\cup L$ and $h(p)\in \Delta _ K$. Holomorphic convexity of $K$ gives $a\in \mathcal O(Y)$ with $|a(p)|>\sup _ K|a|$. Choose polynomials tending uniformly to one on $\Delta _ K$ and to zero on $\Delta _ L$. Their compositions with $h$, multiplied by $a$, converge to $a$ at $p$ and on $K$, and to zero on $L$. A sufficiently good approximation therefore separates $p$ from $K\cup L$. The case $h(p)\in \Delta _ L$ is identical, with $K$ and $L$ interchanged. Thus no point outside $K\cup L$ belongs to its hull. $\square$