# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by E. G. Kwon
Proc. Amer. Math. Soc. 116 (1992), 27-34 Request permission

## Abstract:

Let $B$ be the open unit ball in ${{\mathbf {C}}^n}$ with its boundary $S$. Suppose that $\alpha \geq \tfrac {1}{2}$ and $u(z) = {(1 - {\left | z \right |^2})^{n(1 - \alpha )}}F(z)$ for some $F(z) \in C(\overline B )$. If for every $z \in B$ there corresponds an $r(z):0 < r(z) < 1$ and an automorphism ${\psi _z}$ with ${\psi _z}(0) = z$ such that $u(z) = \frac {1}{{{g_\alpha }(r(z))}}\int _S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,$ then $\tilde \Delta u(z) = - 4{n^2}\alpha (1 - \alpha )u(z),z \in B$. Here $\tilde \Delta$ is the invariant Laplacian and ${g_\alpha }(r)$ is the hypergeometric function $F(n - n\alpha ,n - n\alpha ,n;{r^2})$.
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