Minimization of the conformal radius under circular cutting of a domain

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Let D be a simply connected domain on the complex plane such that 0 ∈ D. For r > 0, let Dr be the connected component of D ∩ z: |z| < r containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(Dr, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local ε-extension Eε(a) of E among all continua E with given logarithmic capacity. Here, Eε(a) = E ∪ z: |z-a| ≤ ε, α ∈ E, ε > 0.

Original languageEnglish
Article number341655
Pages (from-to)2220-2234
Number of pages15
JournalJournal of Mathematical Sciences
Issue number4
StatePublished - 2001


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