TY - JOUR

T1 - A minimal area problem for nonvanishing functions

AU - Barnard, R. W.

AU - Richardson, C.

AU - Solynin, A. Yu

PY - 2007

Y1 - 2007

N2 - The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions f(0) = 1, fʹ(0) = α. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon a priori knowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an explicit solution.

AB - The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions f(0) = 1, fʹ(0) = α. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon a priori knowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an explicit solution.

KW - Minimal area problem

KW - Nonvanishing analytic function

KW - Symmetrization

KW - Typically real function

UR - http://www.scopus.com/inward/record.url?scp=85009729316&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-06-00941-1

DO - 10.1090/S1061-0022-06-00941-1

M3 - Article

AN - SCOPUS:85009729316

SN - 1061-0022

VL - 18

SP - 21

EP - 36

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

IS - 1

ER -