Generalization of a class of nonlinear averaging integral operators

摘要

Let H(U) be the space of all analytic functions in the unit disk U, and let co E denote the convex hull of the set E {is contained in} C. If K {is contained in} H(U) then the operator I : K → H(U) is said to be an averaging operator if I[f](0) = f(0) and I[f](U) {is contained in} co f(U), for all f ∈ K. For a function h ∈ A {is contained in} H(U) we will determine simple sufficient conditions on h such that f(z) < k(z) => I{sub}(h;β,γ)[f](z) < k(z), for all f ∈ M'{sub}(1/β), where I{sub}(h;β,γ)[f](z) = [γ/h{sup}γ(z) ∫ f{sup}β(t)h{sup}(γ-1)(t)h'(t)dt]{sup}(1/β){t from 0 to z} and M'{sub}(1/β) represents the class of l/β-convex functions (not necessarily normalized). As an application, we will give sufficient conditions on h to insure that the operators I{sub}(h;β,γ) are averaging operators on certain subsets of H(U), in order to generalize the result of [5]. In addition, some particular cases of this result obtained for appropriate choices of the function h will also be given.