Accession Number : ADA139256

Title :   On a Theorem of Szegoe on Univalent Convex Maps of the Unit Circle.

Descriptive Note : Technical summary rept.,


Personal Author(s) : Schoenberg,I J

PDF Url : ADA139256

Report Date : Feb 1984

Pagination or Media Count : 18

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.

Descriptors :   *Mapping, *Circles, *Convex sets, Theorems, Geometry, Polynomials, Convergence, Curvature, Conformal mapping

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE