The moduli of certain families of curves and the range of $f(\zeta_0)$ in the class of univalent functions with real coefficients

By simultaneously considering two moduli problems for pairs of homotopic classes of curves a complete solution is obtained of the problem of the range of functions of the class $S_R$, where $S_R$ is the class of functions in $S$ with real coefficients, at a fixed point $\zeta_0$ of the disk $|\zeta|<1$, and $\min|f(\zeta_0)|$ in the class $S_R$ is also found. Partial results in this problem were obtained earlier by J. Jenkins and V. V. Chernikov.