Geodesic
A problem of Salem and Zygmund on the smoothness of an analytic function that generated a Peano curve
Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 485-497 
A. S. Belov. A problem of Salem and Zygmund on the smoothness of an analytic function that generated a Peano curve. Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 485-497. http://geodesic.mathdoc.fr/item/SM_1991_70_2_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\gamma_0=1/2$.

[1] Salem R., Zygmund A., “Lacunary power series and Peano curves”, Duke Math. J., 12:4 (1945), 569–578 | DOI | MR | Zbl

[2] Kahane J.-P., Weiss G., Weiss M., “On lacunary power series”, Arkiv för Matem., 5:1 (1963), 1–26 | DOI | MR | Zbl

[3] Belov A. S., “Stepennye ryady i krivye Peano”, Izv. AN SSSR. Ser. matem., 49:4 (1985), 675–704 | MR | Zbl