Denjoy Domains and BMOA

A Denjoy domain is a plane domain whose complement is a closed subset $E$ of the extended real line $\bar{R}$ containing $\mathrm{\infty }$ : such a domain is called Carleson-homogeneous if there exists $C>0$ such that for all $z\in E$ and $r>0$, one has $\mathrm{\mid }E\cap [z-r,z+r]\mathrm{\mid }\ge Cr$, where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Lebesgue measure on the line. We prove that if $U=\bar{\mathbb{C}}\mathrm{\backslash }K$ is a Carleson-homogeneous Denjoy domain then, if $f$ stands for one of its universal coverings, $\log f^{\mathrm{\prime }}\in BMOA\mathrm{.}$ In order to prove this result, we develop ideas from [On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups, Ann. Fenn. Math. 46(2021),67-77] leading to a general theorem about planar domains giving sufficient conditions ensuring that $\log f^{\mathrm{\prime }}\in BMOA$ for any universal covering $f\mathrm{.}$