Weighted composition operators on the class of subordinate functions (Q1627610)

The authors study the properties of weighted composition operator on the classes \({\mathcal P}_{\alpha}\) defined due to subordination of analytic functions in the unit disc \({\mathbb D}\): \[ {\mathcal P}_{\alpha} := \big\{f\in {\mathcal H}({\mathbb D}): f(z) \prec h_{\alpha}(z)\big\},\quad |\alpha| \leq 1,\quad \alpha\not= -1, \] where \(h_{\alpha}(z) = (1 + \alpha z)/(1 + z)\). For a given analytic self-map \(\phi\) of \({\mathbb D}\) and an analytic map \(\psi\) of \({\mathbb D}\), the corresponding weighted composition operator \(C_{\psi,\phi}\) is defined by \[ C_{\psi,\phi}(f) := \psi (\phi\circ f)\;\; {\mathrm{for}} \;\; f\in {\mathcal H}({\mathbb D}). \] The main result is the following. {Theorem}. Let \(\psi = h_{\alpha} \circ \omega\) and let \(\phi, \omega\) be Schwarz functions. Then \(C_{\psi,\phi}\) preserves the class \({\mathcal P}_{\alpha}\) if and only if \[ 2 Q(\omega) |\phi| < (|1 - |\omega|^2) + P(\omega) |\phi|^2\;\text{ on }{\mathbb D}, \] where \(P(\omega) = |\alpha \omega|^2 - |1 + (\alpha - 1)\omega|^2\), \(Q(\omega) = |(\alpha - 1) |\omega|^2 + \overline{\omega} - \alpha \omega|\). Special cases are considered. Examples are given. The results generalize the recent results of the paper [\textit{I. Arévalo} et al., Monatsh. Math. 187, No. 3, 459--477 (2018; Zbl 1402.30015)].