Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access December 16, 2022

Paatero’s V(k) space II

  • Valentin V. Andreev EMAIL logo , Miron B. Bekker and Joseph A. Cima
From the journal Concrete Operators


In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

MSC 2010: 30C45; 30C15; 30H10


[1] Md Firoz Ali, and A. Vasudevarao, On certain families of analytic functions in the Hornich space, Comput. Methods Func. Theory 18 (2018).10.1007/s40315-018-0244-4Search in Google Scholar

[2] Valentin V. Andreev, Miron B. Bekker, and Joseph A. Cima, Paatero’s V(k) space and a claim by Pinchuk, Proc. Amer. Math. Soc. 150 (2022), 1711-1717.10.1090/proc/15790Search in Google Scholar

[3] Valentin V. Andreev, Miron B. Bekker, and Joseph A. Cima, Paatero’s classes V(k) as subsets of Hornich space, Comput. Methods Func. Theory (2022).10.1007/s40315-022-00472-2Search in Google Scholar

[4] D. A. Brannan, On functions of bounded boundary rotation I, Proc. Edinburgh Math. Soc. 16 (1968-69), 339–347.10.1017/S001309150001302XSearch in Google Scholar

[5] Joseph A. Cima, The Hornich space and a subspace of BMOA, Irish Math. Soc. Bull. 83 (2019), 9–16.10.33232/BIMS.0083.9.16Search in Google Scholar

[6] J. A. Cima, and J. A. Pfaltzgraff, A Banach space of locally univalent functions, Michigan Math. J. 17 (1970), 321–334.10.1307/mmj/1029000518Search in Google Scholar

[7] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1969.Search in Google Scholar

[8] P. Duren, Theory of Hp-spaces, Academic Press, New York, 1970.Search in Google Scholar

[9] P. Duren, Univalent Functions, Springer, Berlin, Heidelberg, 1983.Search in Google Scholar

[10] John B. Garnett, Bounded Analytic Functions, Academic Press, New York, London, Toronto, 1981.Search in Google Scholar

[11] K. Löwner, Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheiskreises |z| < 1, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden, Ber. Verh. Sächs. Ges. Wiss. Leipzig 69 (1917), 89–106.Search in Google Scholar

[12] Javad Mashreghi, Representation Theorems in Hardy Spaces, Cambridge University Press, Cambridge, 2009.10.1017/CBO9780511814525Search in Google Scholar

[13] James W. Noonan, Boundary Behavior of Functions with Bounded Boundary Rotation, J. Math. Anal. Appl. 38 (1972), 721–734.10.1016/0022-247X(72)90079-0Search in Google Scholar

[14] V. Paatero, Über Gebiete van beschränkter Randdrehung, Ann. Acad. Sci. Fenn. Ser. A 37 (1933).Search in Google Scholar

[15] Bernard Pinchuk, The Hardy class of functions of bounded boundary rotation, Proc. Amer. Math. Soc. 38 (1973), 355–360.10.1090/S0002-9939-1973-0313516-7Search in Google Scholar

[16] Ch. Pommerenke, Boundary Behavior od Conformal Maps, Springer, Berlin, Heidelberg, 1992.10.1007/978-3-662-02770-7Search in Google Scholar

[17] Saminathan Ponnusamy, Swadesh Kumar Sahoo, and Toshiyuki Sugawa, Hornich operators on functions of bounded boundary rotations and order α, Comput. Methods Func. Theory 19 (2019), 455–472.10.1007/s40315-019-00276-xSearch in Google Scholar

[18] T. J. Suffridge, On univalent polynomials, J. London Math. Soc. 44 (1969), 496–504.10.1112/jlms/s1-44.1.496Search in Google Scholar

Received: 2021-12-28
Accepted: 2022-07-21
Published Online: 2022-12-16

© 2022 Valentin V. Andreev et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 6.6.2023 from
Scroll to top button