Abstract
In this article we consider Paatero’s classes V(k) of functions of bounded boundary rotation as subsets of the Hornich space \(\mathcal H\). We show that for a fixed \(k\ge 2\) the set V(k) is a closed and convex subset of \(\mathcal H\) and is not compact. We identify the extreme points of V(k) in \(\mathcal H\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
Not applicable.
Code Availability
Not applicable.
References
Andreev, V.V., Bekker, M.B., Cima, J.A.: Paatero’s \(V(k)\) space and the claim of Pinchuk. Proc. Am. Math. Soc. 150, 1711–1717 (2022)
Brannan, D.A., Clunie, J.G., Kirwan, W.E.: On the Coefficient Problem for Functions of Bounded Boundary Rotation. Ann. Acad. Sci. Fenn. Ser. A 523, 3–18 (1973)
Joseph, A.: Cima, On the dual of Hornich’s space. Proc. Am. Math. Soc. 22, 102–103 (1969)
Cima, J.A., Pfaltzgraff, J.A.: A Banach space of locally univalent functions. Mich. Math. J. 17, 321–334 (1970)
Duren, P.L.: Univalent Functions. Springer, New York Berlin (1983)
Hornich, H.: Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen. Monatsh Math. 73, 36–45 (1969)
Loewner, C.: Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises \(|z| < 1\), die durch Funktionen mit nicht verschwindender Ableitung geliefert werden. Ber. Verh. Sächs. Gess. Wiss. Leipzig 69, 89–106 (1917)
James, W.: Noonan, boundary behavior of functions with bounded boundary rotation. J. Math. Anal. Appl. 38, 721–734 (1972)
Paatero, V.: Über Gebiete von beschränkter Randdrehung. Ann. Acad. Sci. Fenn. Ser. A 37 (9), 1–20 (1933)
Ponnusamy, S., Sahoo, S.K., Sugawa, T.: Hornich operations on functions of bounded boundary rotations and order \(\alpha \). Comput. Methods Funct. Theory 19, 455–472 (2019)
Rosenblum, M., Rovnyak, J.: Topics in Hardy Classes and Univalent Functions. Birkhäuser, Basel (1994)
Acknowledgements
The authors are very thankful to the anonymous referees for useful and important comments.
Funding
None reported.
Author information
Authors and Affiliations
Contributions
All authors have contributed equally to this article.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no potential conflict of interest or competing interests.
Ethics approval
Not applicable.
Consent to participate.
Not applicable.
Consent for publication.
Not applicable.
Additional information
Communicated by Raymond Mortini.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Andreev, V.V., Bekker, M.B. & Cima, J.A. Paatero’s Classes V(k) as Subsets of the Hornich Space. Comput. Methods Funct. Theory 23, 689–696 (2023). https://doi.org/10.1007/s40315-022-00472-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-022-00472-2