Stability of geometric properties of integrals and convolutions of meromorphic univalent functions (Q1842476)

Let \(\mathcal A\) denote the class of functions \(f(z) = z + \sum^ \infty_{k = 0} a_ k z^{-k}\) that are analytic in the annulus \(\Delta = \{z : 1 < | z| < \infty\}\) with a simple pole at infinity. For any subclass \({\mathcal M} \subset {\mathcal A}\), let \(\widetilde {\mathcal M} = \{f \in {\mathcal M} : a_ 0 = 0\}\) and \({\mathcal M}_ 0 = \{f \in {\mathcal M} : f(z) \neq 0\) for \(z \in \Delta\}\). Let \(\Sigma\) denote the subset of univalent functions in \(\mathcal A\). Set \(\widetilde{f}(z) = f(z) - a_ 0\). For a fixed point \(z_ 0 \in \Delta\) and complex parameters \(\lambda\) and \(\mu\), the author introduces the integral operators \[ P_ \lambda [f](z) \int^ z_{z_ 0} (f'(t))^ \lambda dt \quad (f \in \Sigma, z\in \Delta)\tag{1} \] and \[ Q_ \mu [f](z) = \int^ z_{z_ 0} (f(t)/t)^ \mu dt \quad (f \in \widetilde{\Sigma}_ 0,\;z \in \Delta). \tag{2} \] The distance functions \(\rho(f,g) = | a_ 0 - b_ 0| + \sum^ \infty_{k = 1} k| a_ k - b_ k|\) and \(\widetilde {\rho} (f,g) = \rho(\widetilde {f}, \widetilde {g})\) are used to define neighborhoods \(N_ \delta(f)\) and \(\widetilde {N}_ \delta (f)\), respectively. Finally, if \({\mathcal M}, {\mathcal N} \subset {\mathcal A}\), then \(T\) is an operator with domain \(D_ T\supset{\mathcal M}\) such that \(T{\mathcal M}\subset{\mathcal N}\), then \(T\) is called \({\mathcal N}\)-stable (with respect to \(\rho\)) at the function \(f_ 0 \in {\mathcal M}\) if there exists a \(\delta > 0\) such that \(T(N_ \delta (f_ 0) \cap D_ T) \subset {\mathcal N}\). In this paper, the author finds sufficient conditions for the operators (1) and (2) to be \(\Sigma\)-stable at functions from \(\Sigma\). In particular, operator (1) turns out to be \(\Sigma\)-stable with respect to \(\widetilde {\rho}\) at the function \(e(z) = z\) for all complex \(\lambda\), and operator (2) turns out to be \(\Sigma\)-stable with respect to \(\rho\) at the function \(e(z) = z\) for all complex \(\mu\). The paper concludes with numerous detailed results about the stability of the convolution \((f*g) (z) = z + \sum^ \infty_{k = 0} a_ k b_ k z^{-k}\) for \(z \in \Delta\).