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Arakelyan's theorem and relations between two harmonic functions
Author(s) -
Anderson J. M.,
Hinkkanen A.
Publication year - 2001
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014510
Subject(s) - mathematics , counterexample , constant (computer programming) , harmonic , harmonic function , function (biology) , pure mathematics , mathematical analysis , discrete mathematics , combinatorics , quantum mechanics , physics , evolutionary biology , computer science , biology , programming language
It is shown that, if h and k are harmonic in ℝ 2 and there exists a positive constant c so that| h + − k + | ⩽ cin ℝ 2 , 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 ℝ 2 .