On sufficient conditions for harmonicity
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- by P. C. Fenton PDF
- Trans. Amer. Math. Soc. 253 (1979), 139-147 Request permission
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 \begin{equation} u(z) = \frac {1} {{2\pi }}\int _0^{2\pi } {u(z + \rho {e^{i\theta }})} d\theta \end{equation} 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
- P. C. Fenton, Functions having the restricted mean value property, J. London Math. Soc. (2) 14 (1976), no. 3, 451–458. MR 437780, DOI 10.1112/jlms/s2-14.3.451 O. D. Kellogg, Foundations of potential theory, Dover, New York, 1953.
- John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0244463
- Lawrence Zalcman, Mean values and differential equations, Israel J. Math. 14 (1973), 339–352. MR 335835, DOI 10.1007/BF02764713
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 253 (1979), 139-147
- MSC: Primary 31A05
- DOI: https://doi.org/10.1090/S0002-9947-1979-0536939-X
- MathSciNet review: 536939