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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 144, Pages 146–148
(Mi znsl5307)
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On the harmonic measure of continua of a fixed diameter
A. Yu. Solynin
Abstract:
Let $\mathcal{E}$ be the family of all continua $E$ in $\bar{U}\setminus\{0\}$, where $U=\{|z|<1\}$, let $U(E)$ be the connected component of $U\setminus E$ containing the point $z=0$, $\omega_E(z_0)=\omega(z_0,E,U(E))$ be the harmonic measure of $E$ relative to the domain $U(E)$ at the point $z_0\in U(E)$. In the paper one answers affirmatively a question raised by B. Rodkin [K.F. Barth, D.A. Branna, and W.K. Hayman, "Research problems in complx analysis,’’ Bull. London Math. Soc.,l6, No. 5, 490–517, 1984]. Namely, one proves that in the family $\mathcal{E}(d_0)$ of continua $E\in\mathcal{E}$, satisfying the condition $\operatorname{diam}E=d_0$, $\quad0<d_0\leq2$, one has the inequality
$$
\omega_E(0)\geq\frac1\pi\arcsin d_0/2,
$$
one indicates all the cases for which equality prevails.
Citation:
A. Yu. Solynin, “On the harmonic measure of continua of a fixed diameter”, Analytical theory of numbers and theory of functions. Part 6, Zap. Nauchn. Sem. LOMI, 144, "Nauka", Leningrad. Otdel., Leningrad, 1985, 146–148
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https://www.mathnet.ru/eng/znsl5307 https://www.mathnet.ru/eng/znsl/v144/p146
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