Close-to-convex Functions And Their Extreme Points In Hornich Space


Let K and C denote, respectively, the classes of normalized convex and close-to-convex univalent functions. Associated with each f∈C there is a collection K (f) of functions g∈K such that f is close-to-convex with respect to g. A characterization of K (f) is given in terms of the radial limits of arg {z f' (z)}, and necessary and sufficient conditions are obtained on f for K (f) to be a singleton. It is shown that for each g∈K there is an f∈C such that K (f)={g}. Further, a characterization is given of those functions f for which K (f) consists only of the half-plane mapping, gβ(z)=z/(1-ze-iβ). These results are used to determine the extreme points of C in the linear space introduced by H. Hornich (Mh. Math. 73, 36-45 (1969)). It is shown that if f is an extreme point of C, then K (f)={gβ} for some β. Finally, a geometric description is given of those functions f∈C for which gβ∈K (f). © 1978 Springer-Verlag.


Mathematics and Statistics

International Standard Serial Number (ISSN)

1436-5081; 0026-9255

Document Type

Article - Journal

Document Version


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Publication Date

01 Sep 1978