Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Conditions on the logarithmic derivative of a function implying boundedness


Authors: T. H. MacGregor and F. Rønning
Journal: Trans. Amer. Math. Soc. 347 (1995), 2245-2254
MSC: Primary 30C45
MathSciNet review: 1277126
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain $ \Omega $. We obtain conditions on $ \Omega $ which imply that the functions are bounded and that their first derivatives belong to $ {H^p}$ for some $ p \geqslant 1$. For certain domains $ \Omega $ the sufficient conditions that we give are also, in some sense, necessary. Examples of domains to which the results apply are given.


References [Enhancements On Off] (What's this?)

  • [1] D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431–443. MR 0251208
  • [2] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • [3] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [4] M. A. Evgrafov, Analytic functions, Dover Publications, Inc., New York, 1978. Translated from the Russian; Reprint of the 1966 original English translation; Edited and with a foreword by Bernard R. Gelbaum. MR 522338
  • [5] R. A. Hibschweiler and T. H. MacGregor, Univalent functions with restricted growth and Cauchy-Stieltjes integrals, Complex Variables Theory Appl. 15 (1990), no. 1, 53–63. MR 1055939
  • [6] Frode Rønning, Integral representations of bounded starlike functions, Ann. Polon. Math. 60 (1995), no. 3, 289–297. MR 1316495

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30C45

Retrieve articles in all journals with MSC: 30C45


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1277126-9
Article copyright: © Copyright 1995 American Mathematical Society