NYJM Logo

New York Journal of Mathematics
Volume 26 (2020), 28-36

  

Horst Alzer and Man Kam Kwong

On two trigonometric inequalities of Askey and Steinig

view    print


Published: January 1, 2020.
Keywords: Trigonometric sums, inequalities, best possible constant.
Subject: 26D05, 26D15, 33B10.

Abstract
We prove that the inequality
(5/8)cos(x/4)+∑{[cos((k+1/4)x)]/[k+1]:1 ≤ k ≤ n} ≥ 0
as well as its companion, obtained by replacing "cos" by "sin", hold for all n ≥ 1 and x ∈ (0,2π). In both cases, the constant factor 5/8 is sharp. This refines a result of Askey and Steinig, who proved the inequalities with the factor 1 instead of 5/8.

Acknowledgements

N/A


Author information

Horst Alzer:
Morsbacher Strasse 10
51545 Waldbröl, Germany

h.alzer@gmx.de

Man Kam Kwong:
Department of Mathematics
The Hong Kong Polytechnic University
Hunghom, Hong Kong

mankwong@connect.polyu.hk