The problem of conformal transformations of a circle into nonoverlapping regions

Let $a$, $a\ne0$, $a\ne\infty$, be a fixed point in the $z$-plane, $\mathfrak M (a,0,\infty)$, the class of all systems $\{f_k(\zeta)\}_1^3$ of functions $z=f_k(\zeta)$, $k=1,2,3$, of which the first two map conformally and in a single-sheeted manner the circle $|\zeta|<1$, and the third maps in a similar manner the region $|\zeta|>1$, into pair-wise nonintersecting regions $B_k$, $k=1,2,3$, containing the points $a,0$, and $\infty$, respectively, so that $f_1(0)=a$, $f_2(0)=0$ and $f_3(\infty)=\infty$. The region of values $\mathscr E(a,0,\infty)$ of the system $M(|f_1'(0)|,|f_2'(0)|,1/|f_3'(0)|)$ in the class $\mathfrak M(a,0,\infty)$ is determined.