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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On sufficient conditions for harmonicity

Author: P. C. Fenton
Journal: Trans. Amer. Math. Soc. 253 (1979), 139-147
MSC: Primary 31A05
MathSciNet review: 536939
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Abstract: Suppose that u is continuous in the plane and that given any complex number z there is a number $ \rho = \rho (z) > 0$ such that

$\displaystyle u(z) = \frac{1} {{2\pi }}\int_0^{2\pi } {u(z + \rho {e^{i\theta }})} d\theta$ (1)

The main result is: if u possesses a harmonic majorant and $ \rho (z)$ is continuous and satisfies a further condition (which may not be omitted) then u is harmonic. Another result in the same vein is proved.

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PII: S 0002-9947(1979)0536939-X
Article copyright: © Copyright 1979 American Mathematical Society