An Extremal Problem for the Hyperbolic Metric on Denjoy Domains | Zendy

Dimitrios Betsakos | Zendy

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An Extremal Problem for the Hyperbolic Metric on Denjoy Domains

Author(s) -

Dimitrios Betsakos

Publication year - 2009

Publication title -

computational methods and function theory

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.627

H-Index - 16

eISSN - 2195-3724

pISSN - 1617-9447

DOI - 10.1007/bf03321754

Subject(s) - mathematics , infimum and supremum , domain (mathematical analysis) , boundary (topology) , complex plane , metric (unit) , hyperbolic geometry , mathematical analysis , interval (graph theory) , plane (geometry) , pure mathematics , combinatorics , geometry , differential geometry , operations management , economics

Suppose that › is a domain in the extended complex plane and assume › contains the origin and that the boundary of › lies on the interval (¡1;1) and has total length 2m, 0 < m < 1. We study the problem of flnding the inflmum of the density of the hyperbolic metric ‚(0;›) at the origin among all such domains ›. We show that if m is su-ciently large, the inflmum is attained uniquely for the doubly connected domain which is symmetric in the imaginary axis. This result improves an estimate of A.Yu. Solynin. We also show that for su-ciently small m the above domain is not any more extremal.

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