Exercise 23, § 3

Let Ω be the set of $(z_1, z_2)\in \mathbf{C}^2$ such that $|z_1|\leqslant 1,|z_2|\leqslant 1$. Let $\alpha \in [0,1]$ and $\Omega_{\alpha}$ be the set of $(z_1, z_2)\in \mathbf{C}^2$ such that

(a) The complements of Ω and $\Omega_{\alpha}$ in $\mathbf{C}^2$ are connected.

(b) Ω is the polynomially convex envelope of $\Omega_{\alpha}$. (If $(z_1^0, z_2^0)\in \Omega$ and if $P\in \mathbf{C}[z_1, z_2]$, there exists $z\in \mathbf{C}$ such that $|z|= 1$ and $|P(z_1^0, z_2^0)|\leqslant |P(z, z_2^0)|$; now $|P(z, z_2^0)|\leqslant$ sup$_{(z_1,z_2)\in\Omega_{\alpha}}|P(z_1, z_2)|.$)