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Proceedings of the American Mathematical Society

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Extremal properties of real axially symmetric harmonic functions in $ E\sp{3}$


Author: Peter A. McCoy
Journal: Proc. Amer. Math. Soc. 67 (1977), 248-252
MSC: Primary 31B05; Secondary 35C10
MathSciNet review: 0457754
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Abstract: The set $ \mathcal{H}$ consists of all real harmonic functions defined by $ U(r,\theta ) = \Sigma _{k = 0}^\infty {a_k}{r^k}{P_k}(\cos \theta )$ which are regular in $ \Sigma $, the open unit sphere about the origin $ {E^3}$. Two problems arise concerning $ \mathcal{H}$ and a subset $ {\mathcal{H}_\ast}$ whose members U have the first $ n + 1$ coefficients $ {a_0}, \ldots ,{a_n}$ specified. (1) For $ U \in \mathcal{H}$, determine $ I(U) = \inf \{ U(r,\theta )\vert(r,\theta ) \in \Sigma \} $ as the limit of a monotone sequence of constants $ \{ {\lambda _k}({a_0}, \ldots ,{a_k})\} _{k = 0}^\infty $ which can be computed algebraically. (2) Find $ {U_0} \in {\mathcal{H}_\ast}$ and the constant

$\displaystyle {\lambda _n}({a_0}, \ldots ,{a_n}) = {\operatorname{Sup}}\left\{ {I(U)\vert U \in {\mathcal{H}_\ast}} \right\} = I({U_0}).$

These are answered by means of the Bergman Integral Operator Method and applications of the Methods of Ascent and Descent to the classical Carathéodory-Fejér problem regarding extremal properties of harmonic functions in $ {E^2}$.

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DOI: https://doi.org/10.1090/S0002-9939-1977-0457754-5
Keywords: Bergman and Gilbert's integral operators, extremal properties of harmonic functions, Carathéodory-Fejér theorems
Article copyright: © Copyright 1977 American Mathematical Society