On an extremal property of nonnegative trigonometric polynomials

摘要: Extending an inequality of Egervary and Szasz [2] we prove that for the coefficients any nonnegative trigonometric polynomial \({T_n}\left( x \right) = {a_0}/2 + \sum\nolimits_{k 1}^n {({a_k}\;\cos \;kx {b_x}\sin \;kx) \geq 0,\;x \in [0,2\pi [,} \) $$ - {a_0}\cos \frac{\pi }{{2\left[ {\frac{{n k}}{{l k}}} \right] 3}} \leq \operatorname{Re} \;\left( {\left( {{a_k} i{b_k}} \right)\gamma \left( {{a_l} i{b_l}} \right)\delta } 3}}$$ holds, where n ≥ 2, k,l are fixed natural numbers with $$\frac{{n 1}}{2} k < l n$$ and γ, δ complex absolute value 1. In particular have $${\left( {a_k^2 b_k^2} \right)^{1/2}} {a_l^2 b_l^2} \;{a_0}\;\cos 3}}.$$ The cases equality discussed too.