Regularity of certain small subharmonic functions

Author:
P. C. Fenton

Journal:
Trans. Amer. Math. Soc. **262** (1980), 473-486

MSC:
Primary 31A05

DOI:
https://doi.org/10.1090/S0002-9947-1980-0586729-5

MathSciNet review:
586729

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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.

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Article copyright:
© Copyright 1980
American Mathematical Society