Harmonic measure of continua with a fixed diameter

Abstract

Let ɛ be the family of all continua E in$\bar U\backslash \left\{ 0 \right\}$, where U={¦z¦ < 1}, let U(E) be the connected component of UE containing the point z=0, let w_E(z₀=w(z₀, E, U(E)) be the harmonic measure of E relative to the domain U(E) at the point z₀ ∃ 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 ɛ(d₀) of continua E ∃ ɛ, satisfying the condition diam E=d₀,o<d₀⩽2, one has the inequality$\omega _{\rm E} (0) \geqslant \frac{1}{\pi }$ arcsin d₀/2, one indicates all the cases for which equality prevails.