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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The infimum of small subharmonic functions


Author: P. C. Fenton
Journal: Proc. Amer. Math. Soc. 78 (1980), 43-47
MSC: Primary 31A05
MathSciNet review: 548081
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Abstract: Suppose that u is subharmonic in the plane and that, for some $ p > 1,{\underline {\lim } _{r \to \infty }}B(r)/{(\log r)^p} = \sigma < \infty $. It is shown that, given $ \varepsilon > 0$,

$\displaystyle A(r) > B(r) - (\sigma + \varepsilon )\operatorname{Re} \{ {(\log r)^p} - {(\log r + i\pi )^p}\} $

for r outside an exceptional set E, where

$\displaystyle \mathop {\underline {\lim } }\limits_{x \to \infty } \;\frac{1}{{... ...og t)}^{p - 2}}}}{t}\;dt\; \leqslant \frac{\sigma }{{\sigma + \varepsilon }}.} $


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

  • [1] P. D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445–495. MR 0139741
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  • [3] W. K. Hayman, The minimum modulus of large integral functions, Proc. London Math. Soc. (3) 2 (1952), 469–512. MR 0056083
  • [4] Bo Kjellberg, On the minimum modulus of entire functions of lower order less than one, Math. Scand. 8 (1960), 189–197. MR 0125967

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DOI: https://doi.org/10.1090/S0002-9939-1980-0548081-6
Article copyright: © Copyright 1980 American Mathematical Society