Arakelyan's theorem and relations between two harmonic functions

J. M. Anderson; A. Hinkkanen

doi:10.1112/S0025579300014510

Arakelyan's theorem and relations between two harmonic functions

J. M. Anderson, A. Hinkkanen

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It is shown that, if h and k are harmonic in ℝ² and there exists a positive constant c so that |h⁺ - k⁺|≤c in ℝ², where h⁺ = max {h, 0), then it need not follow that h - k is identically a constant. The necessary counterexample is obtained by applying Arakelyan's theorem on approximation by an entire function in certain regions in ℝ².

Original language	English (US)
Pages (from-to)	301-304
Number of pages	4
Journal	Mathematika
Volume	48
Issue number	1-2
DOIs	https://doi.org/10.1112/S0025579300014510
State	Published - 2001

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General Mathematics

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abstract = "It is shown that, if h and k are harmonic in ℝ2 and there exists a positive constant c so that |h+ - k+|≤c in ℝ2, where h+ = max {h, 0), then it need not follow that h - k is identically a constant. The necessary counterexample is obtained by applying Arakelyan's theorem on approximation by an entire function in certain regions in ℝ2.",

author = "Anderson, {J. M.} and A. Hinkkanen",

note = "Funding Information: Acknowledgements. This research was supported by the Volkswagen-Stiftung (RiP-programme at Oberwolfach). The research of the second author has been supported in part by the U.S. National Science Foundation grant DMS-9970281.",

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AU - Hinkkanen, A.

N1 - Funding Information: Acknowledgements. This research was supported by the Volkswagen-Stiftung (RiP-programme at Oberwolfach). The research of the second author has been supported in part by the U.S. National Science Foundation grant DMS-9970281.

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N2 - It is shown that, if h and k are harmonic in ℝ2 and there exists a positive constant c so that |h+ - k+|≤c in ℝ2, where h+ = max {h, 0), then it need not follow that h - k is identically a constant. The necessary counterexample is obtained by applying Arakelyan's theorem on approximation by an entire function in certain regions in ℝ2.

AB - It is shown that, if h and k are harmonic in ℝ2 and there exists a positive constant c so that |h+ - k+|≤c in ℝ2, where h+ = max {h, 0), then it need not follow that h - k is identically a constant. The necessary counterexample is obtained by applying Arakelyan's theorem on approximation by an entire function in certain regions in ℝ2.

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Arakelyan's theorem and relations between two harmonic functions

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