Radially distributed values and normal families

Let $L_0$ and $L_1$ be two distinct rays emanating from the origin and let $\mathcal{F}$ be the family of all functions holomorphic in the unit disk $\mathbb{D}$ for which all zeros lie on $L_0$ while all $1$-points lie on $L_1$. It is shown that $\mathcal{F}$ is normal in ${\mathbb D}\backslash{0}$\mathrm{.}Thecasewhere$L_0$isthepositiverealaxisand$L_1$ is the negative real axis is studied in more detail.