Univalence conditions of an integral operator on the exterior unit disk

Into this article, we consider the subclasses $ V_j, V_{j,\mu}$ and $\sum_j(p),$ with $j=2,3,...$, and generalize univalence conditions for the integral operator $ G_{\alpha_i, \beta}$ of the analytic functions $g$ in the exterior unit disk. We want to see if some univalent conditions for analytic functions obtained on the interior unit disk can be extended on the exterior unit disk, so we make use of the usual transformation $$g(z)=\frac{1}{f(\frac{1}{z})} .$$