Regularity of certain small subharmonic functions

Abstract:

Suppose that u is subharmonic in the plane and that ${\underline {\lim } _{r \to \infty }}B(r)/{(\log r)^{2 }} = \sigma < \infty$. It is known that, given $\varepsilon > 0$, there are arbitrarily large values of r such that $A(r) > B(r) - (\sigma + \varepsilon ){\pi ^2}$. The following result is proved. Let u be subharmonic and let $\sigma$ be any positive number. Then either $A(r) > B(r) - {\pi ^2}\sigma$ for certain arbitrarily large values of r or, if this is false, then \[ \lim \limits _{r \to \infty } \left ( {B\left ( r \right ) - \sigma {{\left ( {\log r} \right )}^2}} \right )/\log r\] exists and is either $+ \infty$ or finite.