On the coefficients of concave univalent functions

Abstract

Let D denote the open unit disc and f : D → C be meromorphic and injective in D. We assume that f is holomorphic at zero and has the expansionEspecially, we consider f that map D onto a domain whose complement with respect to C is convex. We call these functions concave univalent functions and denote the set of these functions by Co. We prove that the sharp inequalities |an| ≥ 1, n ∈ N, are valid for all concave univalent functions. Furthermore, we consider those concave univalent functions which have their pole at a point p ∈ (0, 1) and determine the precise domain of variability for the coefficients a2 and a3 for these classes of functions. Equality is attained for one n ∈ N \ {1} if and only if g(z)The present article is devoted mainly to the proof of a perhaps surprising appendage to this result on convex functions. We consider functions f : D → C that are meromorphic and univalent in D, holomorphic at zero and have the expansion

Sharp inequalities for the coefficients of concave schlicht functions

“…In [1] it was conjectured that for concave schlicht functions the inequalities |a n (f )| ≥ 1, n ≥ 2, are valid. This conjecture was proved in [2] and we also conjectured in the same article that the inequalities…”

“…The case n = 3 of (4) was proved in [2] and the cases n = 4 and n = 5 in [9]. According to the above observation this implies that (1) has been proved for n = 2, 3, 4, 5.…”

“…Concerning the history of the conjecture (1) we mention that (1) is the limiting case of another conjecture from [2] that seems to be more difficult to prove than (1). That conjecture concerns functions f : D → C meromorphic and injective in D such that C \ f (D) is convex, f has a simple pole at the point p ∈ (0, 1) and a normalized expansion…”

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.…”

T-Neighborhoods in Various Classes of Analytic Functions

“…It is well known [1], that if f ∈ Co, then |a n | ≥ 1 for all n > 1 and equality holds if and only if f (z) = z/(1 − µz), |µ| = 1 (see [1,3]). The authors in [2] considered the class Co(p) ⊂ Co consisting of all concave functions that have a pole at the point p and are analytic in |z| < |p|.…”