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A problem on the Riesz-Dunford operator calculus and convex univalent functions

Published online by Cambridge University Press:  18 May 2009

J. S. Hwang
J. S. Hwang
Affiliation:
Institute of Mathematics, Academia Sinica Taipei, Taiwan.
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In his paper [3], Ky Fan asked whether if f is a convex univalent function in the unit disk, with f(0) = 0 and f'(0) = 1, then is it true that the set of f(A) is a convex set of operators, when A runs through all proper contractions on a Hilbert space? We answer this question in the negative.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 2 , July 1983 , pp. 129 - 130
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.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
2.Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory. (Interscience, 1958).Google Scholar
3.Fan, Ky, Analytic functions of a proper contraction, Math. Z. 160, (1978) 275–290.CrossRefGoogle Scholar
4.Hille, E., Analytic function theory II. (Ginn, 1962).Google Scholar