Averaging operators and a generalized Robinson differential inequality (Q2367600)

Let \(\text{co} E\) denote the convex hull of a set \(E\) in \(C\), let \(H\) be the class of analytic functions defined in the unit disc \(U\), and denote the subordination of functions \(f,g \in H\) by \(f \prec g\). This paper deals with the theory of averaging operators defined on a set \(K \subset H\). These are operators \(I:K \to H\) that satisfy \(I[f](0)=f(0)\) and \(I[f](U) \subset \text{co} f(U)\) for all \(f\in K\). Conditions for and examples of such operators are given in this paper. The proofs of these results are dependent on determining dominants of the second order differential subordination \(Az^ 2p''(z)+B(z)zp'(z)+C(z)p(z)+D(z) \prec h(z)\). With suitable conditions on \(A,B,C,D\) and \(h\) the authors show that \(p \prec h\). As an additional application of this differential subordination the authors generalize a differential inequality of \textit{R. M. Robinson} [Trans. Am. Math. Soc. 61, 1-35 (1947; Zbl 0032.15603)].