Nonvanishing Univalent Functions*

The class S of functions g(z) = z + c 2 z 2 + c 3 z 3 + ... analytic and univalent in the unit disk Izr < 1 has been thoroughly studied, and its properties are well known. Our purpose is to investigate another class of functions which, by contrast, seems to have been rather neglected. This is the class S o of functions f ( z ) = 1 + a 1 z + a 2 z Z + . . , analytic, univalent, and nonvanishing in the unit disk, normalized by the condition f(0) = 1. It will become apparent that S O is closely related to the more familiar class S and is in some ways easier to handle. After making a few preliminary observations, we adapt the elementary method of Brickman [23 to obtain information about the extreme points and support points of S o. We then use Schiffer's method of boundary variation to consider a wide class of extremal problems and to study the support points of S o in greater depth. Whereas the geometry of the arcs omitted by extreme points and support points of S is related to the families of linear rays and circles centered at the origin, it turns out that the corresponding geometry for S o is related to the families of ellipses and hyperbolas with loci at 0 and 1. The paper concludes with a detailed study of the specific linear extremal problem rain Re{f(~)}, which provides an interesting family of support points in S 0.