## Conditions on the logarithmic derivative of a function implying boundedness

HTML articles powered by AMS MathViewer

- by T. H. MacGregor and F. Rønning PDF
- Trans. Amer. Math. Soc.
**347**(1995), 2245-2254 Request permission

## 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

- D. A. Brannan and W. E. Kirwan,
*On some classes of bounded univalent functions*, J. London Math. Soc. (2)**1**(1969), 431–443. MR**251208**, DOI 10.1112/jlms/s2-1.1.431 - Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - Peter L. Duren,
*Univalent functions*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR**708494** - 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** - 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**, DOI 10.1080/17476939008814433 - Frode Rønning,
*Integral representations of bounded starlike functions*, Ann. Polon. Math.**60**(1995), no. 3, 289–297. MR**1316495**, DOI 10.4064/ap-60-3-289-297

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**347**(1995), 2245-2254 - MSC: Primary 30C45
- DOI: https://doi.org/10.1090/S0002-9947-1995-1277126-9
- MathSciNet review: 1277126