  We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that
are unbounded near the boundary but can be estimated from above by some
(rapidly increasing) radial weight $w$. Our main result gives some conditions
on $w$ that guarantee the estimate from below on the harmonic function by a
multiple of this weight. In dimension two this reverse estimate was first
obtained by M. Cartwright for the case of the power weights,
$w_p(z)=(1-|z|)^{-p}$ for $p>1$, and then generalized to a wide class of
regular weights by a number of authors.
