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Hyperbolic linear invariance and hyperbolic k-convexity

Part of: Entire and meromorphic functions, and related topics Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Wancang Ma  and
David Minda
Wancang Ma
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
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Abstract

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Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

MSC classification

Secondary: 30C99: None of the above, but in this section 30C25: Covering theorems in conformal mapping theory 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 1 , February 1995 , pp. 73 - 93
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Duren, P. L., Univalent functions (Springer, New York, 1983).Google Scholar
[2]Flinn, B. and Osgood, B., ‘Hyperbolic curvature and conformal mapping’, Bull. London Math. Soc. 18 (1986), 272–276.CrossRefGoogle Scholar
[3]Heins, M., Selected topics in the classical theory of functions of a complex variable (Holt, Rinehart and Winston, New York, 1962).Google Scholar
[4]Jørgensen, V., ‘On an inequality for the hyperbolic measure and its applications to the theory of functions’, Math. Scand. 4 (1956), 113–124.CrossRefGoogle Scholar
[5]Ma, W., Mejia, D. and Minda, D., ‘Distortion theorems for hyperbolically and spherically k-convex functions’, in: Proc. of an International Conference on New Trends in Geometric Function Theory and Applications (eds. Parvatham, R. and Ponnusamy, S.) (World Scientific Publishing Co., Singapore, 1991) pp. 46–54.Google Scholar
[6]Ma, W. and Minda, D., ‘Euclidean linear invariance and uniform local convexity’, J. Austral. Math. Soc. (Ser. A) 52 (1992), 401–418.Google Scholar
[7]Ma, W. and Minda, D., ‘Spherical linear invariance and uniform local spherical convexity’, in: Current Topics in Analytic Function Theory (eds. Srivastava, H. M. and Owa, S.) (World Scientific Publishing Co., Singapore, 1992) pp. 148–170.Google Scholar
[8]Mejia, D., The hyperbolic metric in k-convex regions (Ph.D. Thesis, University of Cincinnati, 1986).Google Scholar
[9]Mejia, D. and Minda, D., ‘Hyperbolic geometry in hyperbolically k-convex regions’, Rev. Colombiana Mat. 25 (1991), 123–142.Google Scholar
[10]Minda, D., ‘Hyperbolic curvature on Riemann surfaces’, Complex Variables Theory Appl. 12 (1989), 1–8.Google Scholar
[11]Osgood, B., ‘Some properties of f″/f′ and the Poincaré metric’, Indiana Univ. Math. J. 31 (1982), 449–461.CrossRefGoogle Scholar
[12]Overholt, M., Linear problems for the Schwarzian derivative (Ph.D. Thesis, The University of Michigan, 1987).Google Scholar
[13]Pick, G., ‘Über die konforme Abbildung eines Kreises auf em schlichtes und zugleich beschränktes Gebiet’, S.–B. Kaiserl. Akad. Wiss. Wien, Math. — Naturwiss. Kl. 126 (1917), 247–263.Google Scholar
[14]Pommerenke, Ch., ‘Linear-invariante Familien analytischer Funktionen I’, Math. Ann. 155 (1964), 108–154.Google Scholar
[15]Pommerenke, Ch., ‘Linear-invariante Familien analytischer Funktionen II’, Math. Ann. 156 (1964), 226–262.Google Scholar