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Transactions of the American Mathematical Society

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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.

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

  • [1] P. C. Fenton, Functions having the restricted mean value property, J. London Math. Soc. (2) 14 (1976), no. 3, 451–458. MR 0437780
  • [2] O. D. Kellogg, Foundations of potential theory, Dover, New York, 1953.
  • [3] John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0244463
  • [4] Lawrence Zalcman, Mean values and differential equations, Israel J. Math. 14 (1973), 339–352. MR 0335835

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Article copyright: © Copyright 1979 American Mathematical Society