A class of functions that are univalent in an annulus

In the class $F_1$ of functions $f(\zeta)$, regular and univalent in the annulus $K=\{\rho<|\zeta|<1\}$ and satisfying the conditions $|f(\zeta)|<1$ and $f(\zeta)\ne0$ for $\zeta\in K$, $|f(\zeta)|=1$, $|\zeta|=1$, for $f(1)=1$, one finds the set of the values $D(A)=\{f(A):f\in K\}$ for an arbitrary fixed point $A\in K$. One makes use of the method of variations and certain facts from the theory of the moduli of families of curves.