Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that $$ \left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(2\pi - \int_{\Omega} (\Delta^{-} u) dx\right) \int_{\Omega} e^{2u}dx $$ where $\Delta^{-} u = \max(-\Delta u, 0)$. In particular, if $h$ is a harmonic function then
$$ \left(\int_{\partial \Omega} e^h ds\right)^2 \geq 4\pi \int_{\Omega} e^{2h}dx $$
The paper is published in the Annals.
I am looking for an extension of this result on annulus shaped regions in the following sense. Let $\Omega_1, \Omega_2$ be simply connected regions with $\Omega_1 \subset \subset \Omega_2$, and $h$ be a harmonic function on $\overline{\Omega_2 \backslash \Omega_1}$. Does the following inequality hold
$$ \left(\int_{\partial \Omega} e^h ds\right)^2 \geq 4\pi \int_{\Omega \backslash \Omega_1} e^{2h}dx $$ for any $\Omega$ with $\Omega_1 \subset \Omega \subset \Omega_2.$ The result trivially holds if $h=c$ or when $h$ can be extended to a harmonic function on $\Omega_2$, but I wonder if it holds in general.
