Geodesic
A problem on the Riesz-Dunford operator calculus and convex univalent functions
Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 129-130

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Hwang, J. S. A problem on the Riesz-Dunford operator calculus and convex univalent functions. Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 129-130. doi: 10.1017/S0017089500005188
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[1] 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. Google Scholar | DOI

[2] 2.Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory. (Interscience, 1958). Google Scholar

[3] 3.Fan, Ky, Analytic functions of a proper contraction, Math. Z. 160, (1978) 275–290. Google Scholar | DOI

[4] 4.Hille, E., Analytic function theory II. (Ginn, 1962). Google Scholar

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