On a Theorem of Szegoe on Univalent Convex Maps of the Unit Circle.
Abstract
There is a fine interplay between two fundamental notions of geometry: Convexity and Conformal Mapping. The subject belongs to Geometric Function Theory. In 1928 Gabor Szegoes showed that if a power series converges in the unit circle absolute value z < 1 and maps it onto a convex domain, then all its finite sections map the circle absolute z < 1/4 onto convex domains. The present paper shows that Szegoes theorem reduces to a study of the finite sections of the geometric series 1 + 1/4 z + 1/16 z squared + ... = 1/4 to the (n) power ... z to the (n) power. The main tool is a result conjectured in 1958 by Polya and Schoenberg, but only established in 1973 by St. Ruscheweyh and T. Sheil-Small.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1984
- Accession Number
- ADA139256
Entities
People
- Isaac Jacob Schoenberg
Organizations
- University of Wisconsin–Madison