Abstract:We prove: If $$ \frac 12+\sum _{k=1}^{n}a_k(n)\cos (kx)\geq 0 \text { for all } x\in [0,2\pi ), $$ then $$ 1-a_k(n)\geq \frac 12 \frac {k^2}{n^2} \text { for } k=1,...,n. $$ The constant $1/2$ is the best possible.
Keywords: cosine polynomials, inequalities
AMS Subject Classification: 26D05