Concentration of area in half-planes

Barnard, Roger; Richardson, Clint; Solynin, Alexander

doi:10.1090/S0002-9939-05-07775-0

Concentration of area in half-planes
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by Roger W. Barnard, Clint Richardson and Alexander Yu. Solynin PDF

Proc. Amer. Math. Soc. 133 (2005), 2091-2099 Request permission

Abstract:

For the standard class $S$ of normalized univalent functions $f$ analytic in the unit disk $\mathbb {U}$ , we consider a problem on the minimal area of the image $f(\mathbb {U})$ concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.

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