The infimum of small subharmonic functions

Abstract:

Suppose that u is subharmonic in the plane and that, for some $p > 1,{\underline {\lim } _{r \to \infty }}B(r)/{(\log r)^p} = \sigma < \infty$. It is shown that, given $\varepsilon > 0$, \[ A(r) > B(r) - (\sigma + \varepsilon )\operatorname {Re} \{ {(\log r)^p} - {(\log r + i\pi )^p}\} \] for r outside an exceptional set E, where \[ \underline {\lim } \limits _{x \to \infty } \;\frac {1}{{{{(\log r)}^{p - 1}}}}\int _{E \cap [1,r]} {\frac {{(p - 1){{(\log t)}^{p - 2}}}}{t}\;dt\; \leqslant \frac {\sigma }{{\sigma + \varepsilon }}.} \]