Hardy Number of Koenigs Domains: Sharp Estimate

Let $\Omega$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $\Omega$ is greater or equal to $1/2$. That is, every holomorphic function in the unit disc $f \colon \mathbb{D} \to \Omega$ belongs to the Hardy space $H^{p}(\mathbb{D})$ for all $p<1/2$.