On the Bieberbach and Koebe Constants of a Simply Connected Domain

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dc.contributor.authorCarroll, Tom
dc.contributor.authorRugeihyamu, Sylvester E.
dc.date.accessioned2016-09-21T21:15:08Z
dc.date.available2016-09-21T21:15:08Z
dc.date.issued2001-11-01
dc.descriptionFull text can be accessed at http://www.tandfonline.com/doi/abs/10.1080/17476930108815395en_US
dc.description.abstractThe classical Bieberbach coefficient estimate and Koebe 1/4-theorem for univalent functions in the unit disk may be formulated, in terms of a naturally defined Bieberbach constant n(D) and Koebe constant C(D), as n(D)≤2 and C(D) ≥ 1/4 for each simply connected domain D. We show that these inequalities have the same extremal domains, by means of a variation on the classical argument that yields the Bieberbach and Koebe Theorems, and describe how this is related to work of Flinn and Herron and of Pommerenke. Such extremal domains satisfy n(D)C(D)=1/2. An example of a simply connected domain for which n(o:D)C(D)≠1/2 is constructed, thereby proving that equality does not always hold in the inequality n(D)C(D) ≥½, due to Osgood. In passing, we raise an interesting question on the relationship between the second coefficient of a univalent function f and the radius of the largest disk about f(0) that is covered by fen_US
dc.identifier.citationCarroll, T. and Rugeihyamu, S., 2001. On the bieberbach and koebe constants of a simply connected domain. Complex Variables and Elliptic Equations, 46(1), pp.31-49.en_US
dc.identifier.doi10.1080/17476930108815395
dc.identifier.urihttp://hdl.handle.net/20.500.11810/4240
dc.language.isoenen_US
dc.publisherGordon and Breach Science Publishersen_US
dc.subjectUnivalent functionsen_US
dc.subjectKoebe 1/4-Theoremen_US
dc.subjectBieberbach coefficient theoremen_US
dc.titleOn the Bieberbach and Koebe Constants of a Simply Connected Domainen_US
dc.typeJournal Article, Peer Revieweden_US
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