Close-to-convex functions and their extreme points in Hornich space

Abstract

LetK andC denote, respectively, the classes of normalized convex and close-to-convex univalent functions. Associated with eachf∈C there is a collectionK (f) of functionsg∈K such thatf is close-to-convex with respect tog. A characterization ofK (f) is given in terms of the radial limits of arg {z f' (z)}, and necessary and sufficient conditions are obtained onf forK (f) to be a singleton. It is shown that for eachg∈K there is anf∈C such thatK (f)={g}. Further, a characterization is given of those functionsf for whichK (f) consists only of the half-plane mapping,g _β(z)=z/(1-ze ^−iβ). These results are used to determine the extreme points ofC in the linear space introduced byH. Hornich (Mh. Math.73, 36–45 (1969)). It is shown that iff is an extreme point ofC, thenK (f)={g _β} for some β. Finally, a geometric description is given of those functionsf∈C for whichg _β∈K (f).