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

Karl-Joachim Wirths. "A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions." Annales Polonici Mathematici 83.1 (2004): 87-93. <http://eudml.org/doc/280198>.

@article{Karl2004,
abstract = {Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion $f(z) = z + ∑_\{n=2\}^\{∞\} aₙ(f)zⁿ$, |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability of the coefficient aₙ(f), f ∈ Co(p), for n ∈ 2,3,4,5 is determined by the inequality $|aₙ(f) - (1 - p^\{2n+2\})/(p^\{n-1\}(1-p⁴))| ≤ (p²(1 - p^\{2n-2\}))/(p^\{n-1\}(1-p⁴)). $In the said cases, this settles a conjecture from [1]. 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]) $Re(aₙ(f)) ≥ (1 + p^\{2n\})/(p^\{n-1\}(1+p²))$.},
author = {Karl-Joachim Wirths},
journal = {Annales Polonici Mathematici},
keywords = {concave univalent functions; Livingston conjecture},
language = {eng},
number = {1},
pages = {87-93},
title = {A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions},
url = {http://eudml.org/doc/280198},
volume = {83},
year = {2004},
}

TY - JOUR
AU - Karl-Joachim Wirths
TI - A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions
JO - Annales Polonici Mathematici
PY - 2004
VL - 83
IS - 1
SP - 87
EP - 93
AB - Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion $f(z) = z + ∑_{n=2}^{∞} aₙ(f)zⁿ$, |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability of the coefficient aₙ(f), f ∈ Co(p), for n ∈ 2,3,4,5 is determined by the inequality $|aₙ(f) - (1 - p^{2n+2})/(p^{n-1}(1-p⁴))| ≤ (p²(1 - p^{2n-2}))/(p^{n-1}(1-p⁴)). $In the said cases, this settles a conjecture from [1]. 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]) $Re(aₙ(f)) ≥ (1 + p^{2n})/(p^{n-1}(1+p²))$.
LA - eng
KW - concave univalent functions; Livingston conjecture
UR - http://eudml.org/doc/280198
ER -