A class of nonlinear averaging integral operators
Let \(H\) denote the family of analytic functions in the unit disc \(\Delta = \{z : |z |< 1\}\) and let co \(E\) denote the convex hull of a set \(E\) in the complex plane. If \(K \subset H\) then an operator \(I : K \to H\) is said to be an averaging operator if \(I[f] (0) = f(0)\) and \(I[f] (\Delta) \subset co f (\Delta)\) hold for all \(f \in K\). For \(\beta > 1\) and \(\gamma\), a complex number with \(\text{Re} \gamma > 0\), we let \[ E_{\beta, \gamma} \equiv \biggl\{ f \in H : f(0) = 0,\;f'(0) \neq 0,\;\beta \bigl( zf' (z)/f(z) \bigr) + \gamma \prec Q_{\beta + \gamma} \biggr\} \] and for \(\beta = 1\), we let \(E_{1, \gamma} \equiv H\). Here \(Q_c\) denotes the univalent mapping of \(\Delta\) onto the whole complex plane minus the slit along the half-lines \[ \text{Re} w = 0,\;|Im w |\geq N(c) = {\bigl[ |c |(1 + 2 \text{Re} c)^{1/2} + Im c \bigr] \over \text{Re} c}. \] Under the above assumptions, the authors of this paper describe a class of nonlinear integral operators that are also averaging operators given by \[ [I_{\beta, \gamma} f] (z) \equiv \left[ \gamma z^{- \gamma} \int^z_0 f^\beta (t) t^{\gamma - 1} dt \right]^{1/ \beta} \] and further proved that \(I_{\beta, \gamma}\) is an averaging operator on \(E_{\beta, \gamma}\), where all powers in the above integral are principal values. The idea of the proof of this result relies on two lemmas from the subordination theory.
scientific article; zbMATH DE number 902442