Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31A05

Retrieve articles in all journals with MSC: 31A05


Additional Information

Article copyright: © Copyright 1980 American Mathematical Society