On the univalence of an integral on a subclass of meromorphic convex univalent functions (Q1424058)

Let \(f(z)= z+ \sum^\infty_{n=0} a_n z^{-n}\) belong to the well-known class \(\Sigma\) of functions univalent and meromorphic in \(\Delta= \{z:| z|> 1\}\). Given \(z_0\in\Delta\) and \(\lambda\in \mathbb{C}\), the function \[ P_\lambda[f](z)= \int^z_{z_0} (f'(\zeta))^\lambda \,d\zeta \] is meromrphic in \(\Delta\). We say that \(f\in\Sigma\) is convex if the complement of the image domain is convex. The subclass of \(\Sigma\) consisting of the convex functions is denoted by \(\Sigma_K\). It is found a subset of \(\lambda\in\mathbb{C}\) such that \(P_\lambda(\Sigma_K)\subset\Sigma\). The result is a refinement of a sufficient condition for univalency of the operator \(P_\lambda\) obtained earlier by the first author.