Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in \z\ < 1 and normalized so that f′(0) = 0 and f′(0) = 1, and let R(α,β) ⊂ A be the class of functions f such that Re[f′(z) + αz f″(z)] > β, Re α > 0, β < 1. We determine conditions under which (i) f ∈ R(α1,β1), g ∈ R(α2,β2) implies that the convolution f * g of f and g is convex; (ii) f ∈ R(0,β1), g ∈ R(0,β2) implies that f * g is starlike; (iii) f ∈ A such that f′(z)[f(z)/z]μ-1 ≺ 1 + λz, μ > 0, 0 < λ < 1, is starlike, and (iv) f ∈ A such that f′(z) + αzf″(z) ≺ 1 + δz, α > 0, δ > 0, is convex or starlike. Bibliography: 16 titles. © 1998 Plenum Publishing Corporation.