Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T06:40:27.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Coefficients of an analytic function subordination class determined by rotations

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Seok Chan Kim
Seok Chan Kim
Affiliation:
Department of MathematicsChangwon National UniversityChangwon 641-773, Korea
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.

Let A denote the set of all functions analytic in U = {z: |;z| < 1} equipped with the topology of unifrom convergence on compact subsets of U. For F ∈ A define Let s(F) and s(F) denote the closed convex hull of s(F) and the set of extreme points of , respectively. Let R denote the class of all F ∈ A such that = {Fx}: |x| = 1} where Fx = F(xz).

We prove that |An| ≤ |AMN| for all positive integers M and N, and for . We also prove that if , then F is a univelaent halfplane mapping.

MSC classification

Secondary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 2 , April 1996 , pp. 245 - 254
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Abu-Muhanna, Y., ‘H 1 subordination and extreme points’, Proc. Amer. Math. Soc. 95 (1985), 247–251.Google Scholar
[2]Abu-Muhanna, Y., ‘Subordination and extreme points’, Complex Variables Theory Appl. 9 (1987), 91–100.Google Scholar
[3]Abu-Muhanna, Y. and Hallenbeck, D. J., ‘A class of analytic functions with integral representations’, to appear.Google Scholar
[4]Brannan, D. A., Clunie, J. G. and Kirwan, W. E., ‘On the coefficient problem for functions of bounded boundary rotation’, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 523 (1973).Google Scholar
[5]Brickman, L., MacGregor, T. H. and Wilken, D. R., ‘Convex hulls of some classical families of univalent functions’, Trans. Amer. Math. Soc. 156 (1971), 91–107.CrossRefGoogle Scholar
[6]Feng, J., Extreme points and integral means for classes of analytic functions (Ph.D. Thesis, SUNY at Albany, 1974).Google Scholar
[7]Hallenbeck, D. J. and MacGregor, T. H., Linear problems and convexity techniques in geometric function theory, Monographs and Studies in Math. 22 (Pitman, Boston, 1984).Google Scholar
[8]Hallenbeck, D. J., Perera, S. and Wilken, D. R., ‘Subordination, extreme points and support points’, Complex Variables Theory Appl. 11 (1989), 111–124.Google Scholar
[9]Kim, S. C., Properties of the family of analytic functions with subordination class determined by rotations (Ph.D. Thesis, SUNY, Albany, 1991).Google Scholar
[10]Kim, S. C., ‘Analytic function with subordination class determined by rotations’, Complex Variables Theory Appl. 23 (1993), 177–187.Google Scholar