Subclasses of Starlike Functions Subordinate to Convex Functions

Source abstract

Let S denote the class of functions of the form that are analytic and univalent in the unit disk Δ = { z :| z | < 1}, with S*(α) and K(α) designating the subclasses of S that are, respectively, starlike of order a and convex of order α , 0 ≦ α < 1. If f(z) and g(z) are analytic in Δ, we say that f(z) is subordinate to g(z) , written f ≺ g , if there exists a Schwarz function w(z) , w (0) = 0 and | w(z) | < 1 in Δ, such that f(z) = g ( w(z) ). A function f(z) = z + … is said to be in S *[ A, B ] if (1) and in K [ A, B ] if (2)