Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023 pp. 1–9.

On a One-Parameter Class of Cosine Polynomials

Authors: Horst Alzer (1) and Man Kam Kwong (2)
Author institution:(1) Morsbacher Straße 10, 51545 Waldbröl, Germany
(2) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong

Summary:

We prove: Let $a\geq 0$ be a real number. For any integer $n\geq 2$ and any real $x\in (0,\pi)$, we have $$ 1+\cos(x)+\sum_{k=2}^n \frac{\cos(kx)}{k+a} >\frac{1}{(a+2)(a+3)}. $$ The lower bound is sharp. This extends a result of Brown and Koumandos, who proved the inequality for the special case $a=0$.

2020 Math. Subj. Class. 26D05.

Keywords: Cosine polynomials, inequalities.

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