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A counterexample concerning the maximum and minimum of a subharmonic function


Author: Alexander Fryntov
Journal: Proc. Amer. Math. Soc. 122 (1994), 97-103
MSC: Primary 30D20; Secondary 31A05
DOI: https://doi.org/10.1090/S0002-9939-1994-1189746-5
MathSciNet review: 1189746
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Abstract: For every $ \Delta > 0$ a function u subharmonic in the plane is constructed such that u has the order $ \rho = 1 + \Delta $ and satisfies the condition

$\displaystyle \mathop {\min }\limits_\varphi u(r{e^{i\varphi }})/\mathop {\max }\limits_\varphi u(r{e^{i\varphi }}) \leq - (C + 1)$   for every$\displaystyle \,r > 0,$

where $ C = C(\rho ) > 0$. This example answers a question of W. K. Hayman.

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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1189746-5
Keywords: Subharmonic functions, entire functions, positive harmonic function
Article copyright: © Copyright 1994 American Mathematical Society

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