Area extremal problems for non-vanishing functions
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- by T. Sheil-Small PDF
- Proc. Amer. Math. Soc. 139 (2011), 3231-3245 Request permission
Abstract:
We consider the problem of finding the extremal function $f$which minimises the Bergman space $A^2$norm for the class of non-vanishing functions whose first $n+1$ Taylor coefficients are given.$$We define an analytic function $K$ in terms of $f$ and show that the functions $K$ and $f$ satisfy a certain differential equation. This equation yields a set of relationships between the area moments and the circle moments of $|f|^2$, which in particular shows that the outer part of $f$is a polynomial of degree at most $n$.References
- Dov Aharonov, Catherine Bénéteau, Dmitry Khavinson, and Harold Shapiro, Extremal problems for nonvanishing functions in Bergman spaces, Selected topics in complex analysis, Oper. Theory Adv. Appl., vol. 158, Birkhäuser, Basel, 2005, pp. 59–86. MR 2147588, DOI 10.1007/3-7643-7340-7_{5}
- James G. Caughran, Factorization of analytic functions with $H^{p}$ derivative, Duke Math. J. 36 (1969), 153–158. MR 239095
- T. Sheil-Small, Complex polynomials, Cambridge Studies in Advanced Mathematics, vol. 75, Cambridge University Press, Cambridge, 2002. MR 1962935, DOI 10.1017/CBO9780511543074
Additional Information
- T. Sheil-Small
- Affiliation: Department of Mathematics, University of York, Heslington, York, YO 10 5DD, United Kingdom
- Address at time of publication: P. O. Box 60681, CY8106, Paphos, Cyprus
- Received by editor(s): March 14, 2010
- Received by editor(s) in revised form: July 16, 2010
- Published electronically: April 6, 2011
- Communicated by: Mario Bonk
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3231-3245
- MSC (2010): Primary 32A36; Secondary 30J99
- DOI: https://doi.org/10.1090/S0002-9939-2011-11031-1
- MathSciNet review: 2811279