  Let $\Omega$ be a circular domain, that is, an open disk with finitely many
closed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebra
of all bounded holomorphic functions on $\Omega$, with pointwise operations and
the supremum norm. We show that the topological stable rank of
$H^\infty(\Omega)$ is equal to 2. The proof is based on Suarez's theorem that
the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the
unit disk. We also show that for domains symmetric to the real axis, the Bass
and topological stable ranks of the real symmetric algebra
$H^\infty_\R(\Omega)$ are 2.
