  A Denjoy domain is a plane domain whose complement is a closed subset $E$ of
the extended real line $\bar{R}$ containing $\infty$ : such a domain is called
Carleson-homogeneous if there exists $C>0$ such that for all $z\in E$ and
$r>0$, one has $\vert E\cap [z-r,z+r]\vert\geq Cr$, where $\vert\cdot\vert$ is
the Lebesgue measure on the line. We prove that if $U=\bar{ \mathbb
C}\backslash K$ is a Carleson-homogeneous Denjoy domain then, if $f$ stands for
one of its universal coverings, $\log {f'}\in BMOA.$ 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 $\log {f'}\in
BMOA$ for any universal covering $f.$

Complex Variables