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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Regularity of certain small subharmonic functions
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by P. C. Fenton PDF
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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • 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