Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T23:18:49.587Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTREME POINTS OF INTEGRAL FAMILIES OF ANALYTIC FUNCTIONS

Part of: Geometric function theory Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  14 March 2013

KEIKO DOW  and
D. R. WILKEN
KEIKO DOW*
Affiliation:
Canisius College, 2001 Main Street, Buffalo, NY 14208, USA
D. R. WILKEN
Affiliation:
University At Albany, 1400 Washington Avenue, Albany, NY 12222, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a two-parameter collection of kernel functions integrated against measures on the torus. For specific choices of the parameters many families from classical geometric function theory are included. These families include the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others. The main result introduces a surprising new class of extreme points.

Keywords

extreme points

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 2 , April 2013 , pp. 202 - 221
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Aharonov, D. and Friedland, S., ‘On functions of bounded boundary rotation’, Ann. Acad. Sci. Fenn. A I Math. 585 (1974), 3–18.Google Scholar
Bishop, E., ‘A minimal boundary for function algebras’, Pacific J. Math. 9 (1959), 629–642.CrossRefGoogle Scholar
Gamelin, T., Uniform Algebras (Prentice Hall, Englewood Cliffs, NJ, 1969).Google Scholar
MacGregor, T. H., Brickman, L. and Wilken, D. R., ‘Convex hulls of some classical families of univalent functions’, Trans. Amer. Math. Soc. 156 (1971), 91–107.Google Scholar
MacGregor, T. H. and Wilken, D. R., Handbook of Complex Analysis: Geometric Function Theory, Vol. 1 (North Holland/Elsevier, Amsterdam, 2002), 371–392.CrossRefGoogle Scholar
Perera, S., ‘Support points and extreme points of some classes of analytic functions’, PhD Thesis, State University of New York at Albany, 1983.Google Scholar
Rice, J., ‘Continuous linear functionals on compact and convex integral families of analytic functions’, PhD Thesis, State University of New York at Albany, 1994.Google Scholar