Universally prestarlike functions as convolution multipliers
Let \({\mathcal H}_{1}(\Omega)\) denote the set of functions \(f\) of the form \(f(z) = z + a_2z^2 + \dots\) that are analytic in a domain \(\Omega\) containing the origin. By \({\mathcal R}_{\alpha}\) we denote the class of functions \(f \in {\mathcal H}_{1}(D)\) which are prestarlike of order \(\alpha\) (with \(\alpha \leq 1\)) in the unit disk \(D\). For \(\gamma \in {\mathbb C} \setminus \{0\}\) and \(\rho \in [0,1]\), define \[ \Omega_{\gamma, \rho} = w_{\gamma, \rho}(D),\;\;\;\;w_{\gamma, \rho}(z) = \frac{\gamma z}{1- \rho z},\;z \in D. \] A function \(f \in {\mathcal H}_{1}(\Omega_{\gamma, \rho})\) is called prestarlike of order \(\alpha\) (\(\alpha \leq 1\)) in \(\Omega_{\gamma, \rho}\) if \[ \frac{1}{\gamma}\,f(w_{\gamma, \rho}(z)) \in {\mathcal R}_{\alpha}. \] Let \(\Lambda\) be the slit domain \(\mathbb C \setminus [1, \infty)\). A function \(f \in {\mathcal H}_{1}(\Lambda)\) is called universally prestarlike of order \(\alpha\) (\(\alpha < 1\)) if \(f\) is prestarlike of order \(\alpha\) in all sets \(\Omega_{\gamma, \rho}\) for which \(1 \notin \Omega_{\gamma, \rho}\). The authors prove in particular that the classes of universally prestarlike functions (with \(\alpha\) fixed) are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.
scientific article; zbMATH DE number 5620115