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

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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 $\gamma_0=1/2$. 
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     author = {A. S. Belov},
     title = {A~problem of {Salem} and {Zygmund} on the smoothness of an analytic function that generated {a~Peano} curve},
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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/