## Rudin’s orthogonality problem and the Nevanlinna counting function

HTML articles powered by AMS MathViewer

- by Paul S. Bourdon
- Proc. Amer. Math. Soc.
**125**(1997), 1187-1192 - DOI: https://doi.org/10.1090/S0002-9939-97-03694-0
- PDF | Request permission

## Abstract:

Let $\phi$ be a holomorphic function taking the open unit disk $U$ into itself. We show that the set of nonnegative powers of $\phi$ is orthogonal in $L^2(\partial U)$ if and only if the Nevanlinna counting function of $\phi$, $N_\phi$, is essentially radial. As a corollary, we obtain that the orthogonality of $\{\phi ^n: n=0,1,2,\ldots \}$ for a univalent $\phi$ implies $\phi (z) = \alpha z$ for some constant $\alpha$. We also show that if $\{\phi ^n: n=0,1,2,\ldots \}$ is orthogonal, then the closure of $\phi (U)$ must be a disk.## References

- Sheldon Axler, Paul Bourdon, and Wade Ramey,
*Harmonic function theory*, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR**1184139**, DOI 10.1007/b97238 - Joseph A. Cima, Boris Korenblum, and Michael Stessin, “On Rudin’s Orthogonality and Independence Problem”, preprint.
- Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - Matts Essén and Daniel F. Shea,
*On some questions of uniqueness in the theory of symmetrization*, Ann. Acad. Sci. Fenn. Ser. A I Math.**4**(1979), no. 2, 311–340. MR**565881**, DOI 10.5186/aasfm.1978-79.0404 - M. Essén, D. F. Shea, and C. S. Stanton,
*A value-distribution criterion for the class $L\,\textrm {log}\,L$, and some related questions*, Ann. Inst. Fourier (Grenoble)**35**(1985), no. 4, 127–150 (English, with French summary). MR**812321**, DOI 10.5802/aif.1030 - J. E. Littlewood, “On inequalities in the theory of functions”,
*Proc. London Math. Soc.*(2) 23 (1925), 481–519. - Walter Rudin,
*Real and complex analysis*, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR**0344043** - Joel H. Shapiro,
*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, DOI 10.2307/1971314

## Bibliographic Information

**Paul S. Bourdon**- Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
- Email: pbourdon@wlu.edu
- Received by editor(s): October 27, 1995
- Additional Notes: The author’s research was supported in part by the National Science Foundation (DMS 9401206).
- Communicated by: Theodore W. Gamelin
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 1187-1192 - MSC (1991): Primary 30D50
- DOI: https://doi.org/10.1090/S0002-9939-97-03694-0
- MathSciNet review: 1363413