For functions u, subharmonic in the plane, let
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091510000362/resource/name/S0013091510000362_eqnU1.gif?pub-status=live)
and let N(r,u) be the integrated counting function. Suppose that
is a non-negative non-decreasing convex function of log r for which
for all small r and
, where 1 < ρ < 2, and define
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091510000362/resource/name/S0013091510000362_eqnU2.gif?pub-status=live)
A sharp upper bound is obtained for
and a sharp lower bound is obtained for
.