# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Regularity of certain small subharmonic functionsHTML articles powered by AMS MathViewer

by P. C. Fenton
Trans. Amer. Math. Soc. 262 (1980), 473-486 Request permission

## 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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