On the Phragmen–Lindelof principle for subharmonic functions

We consider subharmonic functions $f(z)$ in a domain $D\subset\mathbb C$ such that $f(z)$ does not exceed some constant $C$ at all points of $\partial D\setminus\zeta$, $\zeta\in\partial D$. Theorems of Phragmen–Lindelof type provide an upper bound (depending on the structure of the domain $D$) for the a priori possible growth of $f(z)$ as $z\to\zeta$ such that functions satisfying this estimate do not exceed $C$ in the whole domain $D$. We obtain a Phragmen–Lindelof type theorem in which the restriction on the possible growth of $f(z)$ as $z\to\zeta$ is expressed in terms of the lower density (with respect to plane Lebesgue measure) of the set $\mathbb C\setminus D$ at the point $\zeta$.