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2004 | 83 | 1 | 87-93
Tytuł artykułu

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

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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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  • Institut für Analysis, TU Braunschweig, 38106 Braunschweig, Germany
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bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-1-10
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