A minimal area problem for nonvanishing functions

The study focuses on determining the smallest area covered by the image of the unit disk for nonvanishing univalent functions that are normalized with specific conditions: f(0) = 1 and fʹ(0) = α. Two distinct approaches are explored to solve this problem.

In the first approach, the problem is simplified using a technique called symmetrization, which transforms it into the realm of typically real functions. This allows for the utilization of a well-known integral representation method to find the solution, provided that certain properties of the extremal function are known in advance.

The second approach, which assumes smoothness conditions, involves variational formulas and leads to a boundary value problem for analytic functions. This problem can be explicitly solved, providing another way to determine the minimal area covered by the image of the unit disk for the specified class of functions.