# zbMATH — the first resource for mathematics

An inequality for the coefficients of a cosine polynomial. (English) Zbl 0833.26012
Summary: We prove: If $${1 \over 2} + \sum^n_{k = 1} a_k(n) \cos (kx) \geq 0$$ for all $$x \in [0,2 \pi)$$, then $$1 - a_k(n) \geq {1 \over 2} {k^2 \over n^2}$$ for $$k = 1, \ldots, n$$. The constant 1/2 is the best possible.
##### MSC:
 26D05 Inequalities for trigonometric functions and polynomials 42A05 Trigonometric polynomials, inequalities, extremal problems
##### Keywords:
inequality; coefficients; cosine polynomial
Full Text: