## Integral means, bounded mean oscillation, and Gel′fer functions

HTML articles powered by AMS MathViewer

- by Daniel Girela
- Proc. Amer. Math. Soc.
**113**(1991), 365-370 - DOI: https://doi.org/10.1090/S0002-9939-1991-1065948-0
- PDF | Request permission

## Abstract:

A Gelfer function $f$ is a holomorphic function in the unit disc $D = \{ z:|z| < 1\}$ such that $f(0) = 1$ and $f(z) + f(w) \ne 0$ for all $z,w$ in $D$. The family $G$ of Gelfer functions contains the family $P$ of holomorphic functions $f$ in $D$ with $f(0) = 1$ and Re $f > 0$ in $D$. Yamashita has recently proved that if $f$ is a Gelfer function then $f \in {H^p},0 < p < 1$ while $\log f \in \operatorname {BMOA}$ and ${\left \| {\log f} \right \|_{\operatorname {BMO}{{\text {A}}_2}}} \leq \pi /\sqrt 2$. In this paper we prove that the function $\lambda (z) = (1 + z)/(1 - z)$ is extremal for a very large class of problems about integral means in the class $G$. This result in particular implies that $G \subset {H^p},0 < p < 1$, and we use it also to obtain a new proof of a generalization of Yamashita’s estimation of the BMOA norm of $\log f,f \in G$.## References

- Albert Baernstein II,
*Integral means, univalent functions and circular symmetrization*, Acta Math.**133**(1974), 139–169. MR**417406**, DOI 10.1007/BF02392144 - Albert Baernstein II,
*Analytic functions of bounded mean oscillation*, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 3–36. MR**623463** - Nikolaos Danikas,
*Über die BMOA-Norm von $\textrm {log}(1-z)$*, Arch. Math. (Basel)**42**(1984), no. 1, 74–75 (German). MR**751474**, DOI 10.1007/BF01198131 - 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** - John B. Garnett,
*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971** - S. Guelfer,
*On the class of regular functions which do not take on any pair of values $w$ and $-w$*, Rec. Math. [Mat. Sbornik] N.S.**19(61)**(1946), 33–46 (Russian, with English summary). MR**0020632** - Daniel Girela,
*Integral means and BMOA-norms of logarithms of univalent functions*, J. London Math. Soc. (2)**33**(1986), no. 1, 117–132. MR**829393**, DOI 10.1112/jlms/s2-33.1.117
A. W. Goodman, - Donald Sarason,
*Function theory on the unit circle*, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR**521811** - Shinji Yamashita,
*Gel′fer functions, integral means, bounded mean oscillation, and univalency*, Trans. Amer. Math. Soc.**321**(1990), no. 1, 245–259. MR**1010891**, DOI 10.1090/S0002-9947-1990-1010891-X

*Univalent functions*, I, II, Mariner, Tampa, 1983.

## Bibliographic Information

- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**113**(1991), 365-370 - MSC: Primary 30C80; Secondary 30D50
- DOI: https://doi.org/10.1090/S0002-9939-1991-1065948-0
- MathSciNet review: 1065948