Harmonic measure of continua with a fixed diameter

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Abstract

Let e{open} be the family of all continua E in {Mathematical expression}, where U={|z| < 1}, let U(E) be the connected component of UE containing the point z=0, let wE(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 e{open}(d0) of continua E ∃ e{open}, satisfying the condition diam E=d0,o<d0≤2, one has the inequality {Mathematical expression} arcsin d0/2, one indicates all the cases for which equality prevails.

Original languageEnglish
Pages (from-to)2140-2142
Number of pages3
JournalJournal of Soviet Mathematics
Volume38
Issue number4
DOIs
StatePublished - Aug 1987

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