Convex meromorphic mappings

Abstract

We study functions f (z) which are meromorphic and univalent in the unit disk with a simple pole at z = p, 0 < p < 1, and which map the unit disk onto a domain whose complement is either convex or is starlike with respect to a point w 0 = 0.

On the coefficients of concave univalent functions

“…These subclasses of Co for p ∈ (0, 1) have been considered by several authors. The functions in Co(p), p ∈ (0, 1), have been called convex meromorphic functions (compare [7] and [9]) and those classes have been denoted by C(p) there. We prefer the present notation which refers to the shape of f (D) rather than to the shape of its complement and avoids confusion with the class of close-to-convex functions.…”

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

“…In fact, at this time Pfaltzgraff and Pinchuk had essentially proved in [7] that the class of functions that satisfy this set of conditions equals Co(p). This fact was recognized by A. E. Livingston in the article [4] published in these Annales. In Livingston's paper, which mainly motivated the research presented here, it was proved that…”

“…On the one hand, the considerations in [5], [7] and [4] concerning (2) together with a little computation show that f ∈ Co(p) if and only if f satisfies (ii) and there exists a function P holomorphic in D such that…”

“…The above inequality was proved for n = 2 in [6] and [2] by different methods and for n = 3 in [1]. A consequence of this inequality is the so called Livingston conjecture (see [4])…”

On Some Problems of James Miller

“…Another related class of our interest is the class S(p) of univalent meromorphic functions f in D with a simple pole at z = p, p ∈ (0, 1), and with the normalization f (z) = z + ∞ n=2 a n (f )z n for |z| < p. If f ∈ S(p) maps D onto a domain whose complement with respect to C is convex, then we call f a concave function with pole at p and the class of these functions is denoted by Co(p). In a recent paper, Avkhadiev and Wirths [2] established the region of variability for a n (f ), n ≥ 2, f ∈ Co(p) and as a consequence two conjectures of Livingston [7] in 1994 and Avkhadiev, Pommerenke and Wirths [1] were settled.…”