Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions

Let $T_r$ be the class of functions $f(z)=z+c_2z^2+\dots$, regular in the disk $|z|<1$, real on the diameter $-1<z<1$, and satisfying the condition $\operatorname{Im}f(z)\cdot\operatorname{Im}z>0$ in the remainder of the disk $|z|<1$. Let $z_f$ be the solution of $f(z)=\alpha f(a)$ on $T_r$, where $\alpha$ is any fixed complex number, $\alpha\ne0$, $\alpha\ne1$, $\alpha$ is any fixed real number, $|\alpha|<1$. We determine the region $D_{T_r}$ of values of the functional $z_f$ on the class $T_r$. Variation formulas for Stieltjes integrals due to G. M. Goluzin are used.