The region of values of the system $\{f(z_1),f(z_2),f(z_3)\}$ on the class of typically real functions

The paper determines the region of values of the system occurring in the title on the class $T$ of functions $f(z)=z+\dotsb$ regular in the unit disk and satisfying the condition $\operatorname{Im}f(z)\cdot\operatorname{Im}z>0$ for $\operatorname{Im}z\ne0$.