Starlike cases of the generalized goodman conjecture

Abstract

We consider functions f that are meromorphic and univalent in the unit disc D with a simple pole at the point p ∈ (0, 1) and normalized by f(0) = f′(0) - 1 = 0. A function g is called subordinated under such a function f, if there exists a function ω holomorphic in D, ω(D) ⊂ D̄, such that g(z) = f(zω(z)), z ∈ D, and we use the abbreviation g ≺ f to indicate this relationship between two functions. We conjectured that for g ≺ f, the inequalities are valid. Here f is as above and the expansion is valid in some neighbourhod of the origin. In the present article, we prove that this is true for two classes of functions f for which C̄\f(D) is starlike. © 2013 Pleiades Publishing, Ltd.