On the change in harmonic measure under symmetrization

V. N. Dubinin

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 267-273

Résumé

Let $D_\alpha$ be the unit disc cut along the segments $l_k=\{z:\arg z=\alpha_k,\ r\leqslant|z|\leqslant1\}$, $k=0,1,\dots,n-1$ ($\alpha=(\alpha_0,\alpha_1,\dots,\alpha_{n-1})$, $0$), and let $\omega_\alpha$ be the harmonic measure of the set $\bigcup\limits_{k=0}^{n-1}l_k$ relative to the region $D_\alpha$ at the point $z=0$. An affirmative solution is given of a problem of A. A. Gonchar: $$ \omega_\alpha\leqslant\omega_{\alpha^*} $$ where $\alpha^*=\bigl(0,\frac{2\pi}n,\dots,\frac{2\pi}n(n-1)\bigr)$. Equality holds only when $D_\alpha$ coincides with $D_{\alpha^*}$ to within a rotation about the origin. The proof is based on a property of certain condensers under dissymmetrization, i.e. under a transformation of symmetric condensers into nonsymmetric ones. Bibliography: 4 titles.