Geodesic
Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions
Matematičeskie zametki, Tome 10 (1971) no. 1, pp. 41-52.

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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 $-1$, 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. 
@article{MZM_1971_10_1_a5,
     author = {L. Kh. Burshtein},
     title = {Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {41--52},
     publisher = {mathdoc},
     volume = {10},
     number = {1},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_1_a5/}
}
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L. Kh. Burshtein. Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions. Matematičeskie zametki, Tome 10 (1971) no. 1, pp. 41-52. http://geodesic.mathdoc.fr/item/MZM_1971_10_1_a5/