Inequalities for two sine polynomials

• Autorzy: Horst Alzer , Stamatis Koumandos
• Języki publikacji: EN

Abstrakty

EN

We prove:
(I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have
$α ≤ ∑_{j=1}^{n-1} 1/(n²-j²) sin(jx) ≤ β$,
with the best possible constant bounds
α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3.

(II) The inequality
$0 < ∑_{j=1}^{n-1} (n²-j²)sin(jx)$
holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

Kategorie tematyczne

• 26D05: Inequalities for trigonometric functions and polynomials
• 42A05: Trigonometric polynomials, inequalities, extremal problems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2006
• Tom: 105
• Numer: 1
• Strony: 127-134

Daty

wydano

2006

Twórcy

autor

Horst Alzer

• Morsbacher Str. 10, D-51545 Waldbröl, Germany

autor

Stamatis Koumandos

• Department of Mathematics and Statistics, The University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Identyfikatory

DOI

10.4064/cm105-1-11