We consider functions of the type $F\left(z\right)=z{\prod }_{j=1}^n{[f_j\left(z\right)/z]}^{a_j}$, where $a_j$ are real numbers and $f_j$ are ${\beta }_j$-strongly close-to-starlike functions of order ${\alpha }_j$. We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.