Coefficient bounds and convolution properties for certain classes of close-to-convex functions

A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional |a₃- μa₂²| for certain classes of univalent functions. In this paper, we consider the class C(φ, ψ) of normalized close-to-convex functions which is defined by using subordination for analytic functions φ and ψ on the unit disk. Our main object is to provide bounds of the quantity a₃ - μa₂² for functions f(z) = z + a₂Z² + a₃z³ + ⋯ in C(φ, ψ) in terms of φ and ψ, where μ is a real constant. We also show that the class C(φ, ψ) is closed under the convolution operation by convex functions, or starlike functions of order 1/2 when φ and ψ satisfy some mild conditions.