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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Value distribution of harmonic polynomials in several real variables.

Author: Morris Marden
Journal: Trans. Amer. Math. Soc. 159 (1971), 137-154
MSC: Primary 31.11
MathSciNet review: 0279323
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Abstract: Using Bergman's integral operator method, the author studies an arbitrary axisymmetric harmonic polynomial $ H(x,\rho )$ in $ {R^3}$ and $ {R^N}$ in relation to its associate polynomial $ h(\zeta )$ in $ C$. His results pertain to the value distributions and critical circles of $ H(x,\rho )$ in certain cones; bounds on the gradient of an $ H(x,\rho )$ assumed bounded in sphere $ {x^2} + {\rho ^2} \leqq 1$; axisymmetric harmonic vectors. Corresponding results are also obtained for axisymmetric harmonic functions $ F(x,\rho )$ with rational associate $ f(\zeta )$.

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Keywords: Axisymmetric harmonic polynomials, Bergman integral method, analytic theory of polynomials, coincidence theorems, axisymmetric harmonic functions, gradient of bounded harmonic polynomials, Bernstein's Theorem, critical circles of axisymmetric harmonic polynomials, axisymmetric harmonic vectors, axisymmetric flow potential, Stoke's stream function
Article copyright: © Copyright 1971 American Mathematical Society