Browsing by Author "Rugeihyamu, Sylvester E."

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    On the Bieberbach and Koebe Constants for Sector Domains and Sector Disks
    (Springer, 2003-12-01) Rugeihyamu, Sylvester E.
    We discuss domain constants related to the classical Bieberbach and Koebe theorems. We find a class of simply connected domains for which the product of these constants behave like extremal domain and gives a better result on Osgood’s inequalities.
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    On the Bieberbach and Koebe Constants of a Simply Connected Domain
    (Gordon and Breach Science Publishers, 2001-11-01) Carroll, Tom; Rugeihyamu, Sylvester E.
    The 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 f
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    On the Koebe and Bieberbach Constants for a Hyperbolic Planar Domain
    (Springer, 2000-12-01) Rugeihyamu, Sylvester E.
    We discuss domain constants related to the classical Bieberbach and Koebe theorems. We find a class of simply connected domains for which the value of these constants is extremal and investigate the relationship between the geometry of a sequence of hyperbolic domains and the associated sequence of domain constants.
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    On the Koebe and Bieberbach Constants for Hyperbolic Domains
    (1999) Rugeihyamu, Sylvester E.