On the Univalence of an Integral on Subclasses of Meromorphic Functions

We study the integral operator $P_\lambda[f](\zeta)=\int_{\zeta_0}^\zeta\bigl(f'(t)\bigr)^\lambda dt$, $|\zeta|>1$, acting on the class $\Sigma$ of functions meromorphic and univalent in the exterior of the unit disk. We refine the ranges of the parameter $\lambda$ for which the operator preserves univalence either on $\Sigma$ or on its subclasses consisting of convex functions. As a consequence, a two-sided estimate is deduced for the separating constant in the sufficient condition for the univalent solvability of exterior inverse boundary-value problems.