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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
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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

$\displaystyle \mathop {\lim }\limits_{r \to \infty } \left( {B\left( r \right)\, - \,\sigma {{\left( {\log \,r} \right)}^2}} \right)/\log \,r$

exists and is either $ + \,\infty $ or finite.

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DOI: https://doi.org/10.1090/S0002-9947-1980-0586729-5
Article copyright: © Copyright 1980 American Mathematical Society