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

Karl-Joachim Wirths

Annales Polonici Mathematici, Tome 83 (2004), p. 87-93 / Harvested from The Polish Digital Mathematics Library

Résumé

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]) $Re(aₙ(f)) ≥ (1 + p^{2n})/(p^{n-1}(1+p²))$.

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Publié le : 2004-01-01

Zbl 1058.30018

EUDML-ID : urn:eudml:doc:280198