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

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 $I(U) = \inf \{ U(r,\theta )|(r,\theta ) \in \Sigma \}$ 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 \[ {\lambda _n}({a_0}, \ldots ,{a_n}) = {\operatorname {Sup}}\left \{ {I(U)|U \in {\mathcal {H}_\ast }} \right \} = I({U_0}).\] 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 ${E^2}$.