A problem of Salem and Zygmund on the smoothness of an analytic function that generated a Peano curve

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 $\gamma_0=1/2$.