Uniqueness Theorems for Some Free Boundary Problems in Univalent Functions

Authors: Huntey J.M.; Jong Moh N.; Tepper D.E.

Source: Complex Variables, Volume 48, Number 7, July 2003 , pp. 607-614(8)

Publisher: Taylor and Francis Ltd

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Abstract:

Let Phi be a bounded positive continuous function in the complex plane . We will consider the following problem. We can find a simply connected region Omega containing the origin such that the Riemann mapping function fOmega mapping the unit disk D onto Omega will satisfy We show that there is a unique solution which is starlike with respect to the origin when Phi satisfies We also show that if in addtion, partPhi/part|w|>0 and -log Phi is subharmonic, then Omega will be a convex region.

Keywords: Free boundary; Univalent function; Reimann mapping; 1991 Mathematics Subject Classifications: Primary; Secondary 31A10

Document Type: Research article

DOI: 10.1080/0278107031000109542

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