Hardy Number of Koenigs Domains: Sharp Estimate

Let $\mathrm{\Omega }$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $\mathrm{\Omega }$ is greater or equal to $1\mathrm{/}2$. That is, every holomorphic function in the unit disc $f\text{\hspace{0.17em}⁣}:\mathbb{D}\to \mathrm{\Omega }$ belongs to the Hardy space $H^p(\mathbb{D})$ for all $p<1\mathrm{/}2$.