Concentration of area in half-planes

Abstract

For the standard class S of normalized analytic functions f univalent on the unit disk U, strict lower bounds on the minimal area of the image f(U) concentrated in any given half-plane are found and explicit formulae given for the extremal maps. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regard 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.