  Let ${H^p}\left( \mathbb{D} \right)$ be the Hardy space of all holomorphic
functions on the unit disk $\mathbb{D}$ with exponent $p>0$. If $D\ne
\mathbb{C}$ is a simply connected domain and $f$ is the Riemann mapping from
$\mathbb{D}$ onto $D$, then the Hardy number of $D$, introduced by Hansen, is
the supremum of all $p$ for which $f \in {H^p}\left( \mathbb{D} \right)$. Comb
domains are a well-studied class of simply connected domains that, in general,
have the form of the entire plane minus an infinite number of vertical rays. In
this paper we study the Hardy number of a class of comb domains with the aid of
the quasi-hyperbolic distance and we establish a necessary and sufficient
condition for the Hardy number of these domains to be equal to infinity.
Applying this condition, we derive several results that show how the mutual
distances and the distribution of the rays affect the finiteness of the Hardy
number. By a result of Burkholder our condition is also necessary and
sufficient for all moments of the exit time of Brownian motion from comb
domains to be infinite.
