On a monotonic trigonometric sum
The main result is the Theorem: For \(n\in\mathbb{N}\), \(\theta\in(0,\pi)\) and \(\alpha\geq 1\), \[ {d\over d\theta} \Biggl(\cos{\theta\over 2} \Biggl(1+ \sum^n_{k=1} {\cos k\theta\over k^\alpha}\Biggr)\Biggr)<0. \] This inequality fails to hold for appropriate \(n\) and \(\theta\) when \(0<\alpha<1\).
scientific article; zbMATH DE number 986869