Averaging Operators and a Generalized Robinson Differential Inequality

Let 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 ∈ H by f ≺ g. This article deals with the theory of averaging operators defined on a set K ⊂ H. These are operators I: K → H that satisfy I[f](0) =f(0) and I[f](U) ⊂ co f(U) for all f ∈ K. Conditions for and examples of such operators are presented. The proofs of these results are dependent on determining dominants of the second-order differential subordination Az2p″(z) + B(z) zp′(z) + C(z) p(z) + D(z) ≺ h(z). With suitable conditions on A, B, C, D, and h the authors show that p ≺ h. As an additional application of this differential subordination the authors generalize a differential inequality of R. M. Robinson