Inequalities for two sine polynomials

Horst Alzer, and Stamatis Koumandos. "Inequalities for two sine polynomials." Colloquium Mathematicae 105.1 (2006): 127-134. <http://eudml.org/doc/284357>.

@article{HorstAlzer2006,
abstract = {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].},
author = {Horst Alzer, Stamatis Koumandos},
journal = {Colloquium Mathematicae},
keywords = {trigonometric polynomials; inequalities},
language = {eng},
number = {1},
pages = {127-134},
title = {Inequalities for two sine polynomials},
url = {http://eudml.org/doc/284357},
volume = {105},
year = {2006},
}

TY - JOUR
AU - Horst Alzer
AU - Stamatis Koumandos
TI - Inequalities for two sine polynomials
JO - Colloquium Mathematicae
PY - 2006
VL - 105
IS - 1
SP - 127
EP - 134
AB - 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].
LA - eng
KW - trigonometric polynomials; inequalities
UR - http://eudml.org/doc/284357
ER -