Concentration of area in half-planes

In the context of a class of normalized univalent functions in the unit disk, researchers are investigating a problem involving the smallest possible area that the image of these functions can occupy within a specified half-plane. This problem is connected to a longstanding question initially posed by A. W. Goodman in 1949, which concerns the minimization of the area covered by analytic univalent functions while adhering to specific geometric constraints.

What makes this problem intriguing is the surprising behavior exhibited by the functions that are candidates for having the smallest possible area, which are constructed based on geometric considerations.