On a trigonometric inequality of Szegő and Schweitzer
Inequalities for trigonometric polynomials have applications in geometric function theory and other fields. The present study has been motivated by \textit{R. Askey} and \textit{J. Fitch} [Publ. Fac. Électrotech. Univ. Belgrade, Sér. Math. Phys. 381--409, 131--134 (1972; Zbl 0254.42001)]. This study proves that the inequality \[ \sum_{\begin{array}{c} k=1 \\ k~odd \end{array} }^{n} (n+1-k)(n+2-k)k \sin kx \geq 0,\] for \(n\in \mathbb{N};\) \(0 \leq x \leq \pi\) and \[ \sum_{\begin{array}{c} k=1 \\ k~even \end{array} }^{n} (n+1-k)(n+2-k)k \sin kx \geq 0,\] for \(2 \leq n\in \mathbb{N};\) \(0 \leq x \leq \frac{1}{2}\) arccos \(-\frac{5}{6}.\) Moreover, the authors use the above inequalities to obtain two one-parameter classes of absolutely monotonic functions and show that these functions satisfy a certain functional inequality, too.
sine polynomials, inequalities, absolutely monotonic
scientific article; zbMATH DE number 7393772
scientific article