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A geometric property of functions harmonic in a disk. (English) Zbl 0758.31001

Denote by \(\Delta\) the unit disc in \(\mathbb{C}\) and by \(\mathbb{T}\) the unit circle, and let \(f\) be an integrable function on \(\mathbb{T}\). For \(z\in\Delta\) and \(e^{it}\in\mathbb{T}\), define \(\ell(z,t)\) to be such that \(z,e^{it}\) and \(e^{i\ell(z,t)}\) are collinear and write \(z=(1- s(z,t))e^{it}+s(z,t)e^{i\ell(z,t)}\). Let \(\ell_ z(t)\) be the corresponding weighted average of \(f(e^{it})\) and \(f(e^{i\ell(z,t)})\). The main theorem states that the average of \(\ell_ z\) on \(\mathbb{T}\) gives the value of the Poisson transform of \(f\) at \(z\). The proof is entirely computational, making use of nice formulas for \(s(z,t)\), \(\ell(z,t)\) and their derivatives in terms of the Poisson kernel \(P_ x(t)\). Unfortunately it does not provide any geometric insight as to why this result should hold. The theorem can also be interpreted as follows: the two-dimensional Poisson integral of \(f\) is average of one-dimensional Poisson integrals. A generalization of this to higher dimensions is announced.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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