Value distribution of harmonic polynomials in several real variables.

Marden, Morris

doi:10.1090/S0002-9947-1971-0279323-1

Value distribution of harmonic polynomials in several real variables.
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Trans. Amer. Math. Soc. 159 (1971), 137-154 Request permission

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 $f(\zeta )$.

References

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