Abstract:
Let $\gamma_0$ denote the supremum of the numbers $\gamma\in(0,1)$ for which there is a function $F\in\operatorname{Lip}\gamma$ on the closed unit disk $D=\{z:|z|\leqslant 1\}$ such that $F$ is analytic inside $D$ and the set $\{F(z):|z|=1\}$ possesses an interior point. In 1945, Salem and Zygmund proved that $\gamma_0\in(0,1/2]$, and asked for the value of $\gamma_0$. It is proved in this paper that $\gamma_0=1/2$.
Citation:
A. S. Belov, “A problem of Salem and Zygmund on the smoothness of an analytic function that generated a Peano curve”, Math. USSR-Sb., 70:2 (1991), 485–497
\Bibitem{Bel90}
\by A.~S.~Belov
\paper A~problem of Salem and Zygmund on the smoothness of an analytic function that generated a~Peano curve
\jour Math. USSR-Sb.
\yr 1991
\vol 70
\issue 2
\pages 485--497
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\crossref{https://doi.org/10.1070/SM1991v070n02ABEH001384}
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Linking options:
https://www.mathnet.ru/eng/sm1208
https://doi.org/10.1070/SM1991v070n02ABEH001384
https://www.mathnet.ru/eng/sm/v181/i8/p1048
This publication is cited in the following 8 articles:
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Krzysztof Barański, Progress in Probability, 70, Fractal Geometry and Stochastics V, 2015, 77
Sophie Grivaux, “A hypercyclic rank one perturbation of a unitary operator”, Math. Nachr, 2012, n/a
Stanislav Shkarin, “A hypercyclic finite rank perturbation of a unitary operator”, Math Ann, 2010
S. Jaffard, S. Nicolay, “Pointwise smoothness of space-filling functions”, Applied and Computational Harmonic Analysis, 26:2 (2009), 181
Baranski K., “On Some Lacunary Power Series”, Mich. Math. J., 54:1 (2006), 65–79
A. Cantón, A. Granados, Ch. Pommerenke, “Borel images and analytic functions”, Michigan Math. J., 52:2 (2004)
S. A. Shkarin, “On a class of continuous nowhere differentiable functions”, Russian Acad. Sci. Izv. Math., 45:2 (1995), 423–432
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