Mapping properties of analytic functions on the disk
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- by Pietro Poggi-Corradini PDF
- Proc. Amer. Math. Soc. 135 (2007), 2893-2898 Request permission
Abstract:
There is a universal constant $0<r_0<1$ with the following property. Suppose that $f$ is an analytic function on the unit disk $\mathbb D$, and suppose that there exists a constant $M>0$ so that the Euclidean area, counting multiplicity, of the portion of $f(\mathbb D)$ which lies over the disk $D(f(0),M)$, centered at $f(0)$ and of radius $M$, is strictly less than the area of $D(f(0),M)$. Then $f$ must send $r_0\overline {\mathbb D}$ into $D(f(0),M)$. This answers a conjecture of Don Marshall.References
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- Donald E. Marshall, A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), no. 1, 131–137. MR 1004727, DOI 10.1007/BF02386365
- Jan Malý, David Swanson, and William P. Ziemer, The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355 (2003), no. 2, 477–492. MR 1932709, DOI 10.1090/S0002-9947-02-03091-X
- Makoto Ohtsuka. Dirichlet problem, extremal length, and prime ends. Van Nostrand, 1970.
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
Additional Information
- Pietro Poggi-Corradini
- Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506
- Email: pietro@math.ksu.edu
- Received by editor(s): January 3, 2006
- Received by editor(s) in revised form: June 1, 2006
- Published electronically: May 8, 2007
- Communicated by: Juha M. Heinonen
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 2893-2898
- MSC (2000): Primary 30C55
- DOI: https://doi.org/10.1090/S0002-9939-07-08823-5
- MathSciNet review: 2317966