Some inequalities for cosine sums (Q2720288)

The main results proved are the following.NEWLINENEWLINENEWLINETheorem 1. For all integers \(n\geq 1\) and \(0<\theta< \pi\), we have NEWLINE\[NEWLINE{41\over 96}+ \sum^n_{k= 1} {\cos k\theta\over k+ 1}\geq 0NEWLINE\]NEWLINE and the constant \(41/96\) is best possible.NEWLINENEWLINENEWLINETheorem 2. For all integers \(n\geq 1\) and \(0< \theta<\pi\), we have NEWLINE\[NEWLINE\sum^n_{k= 1} {\cos k\theta\over k+ 1}< \text{Ci}\Biggl({\pi\over 2}\Biggr)+ \sum^\infty_{k= 1} {\cos k\theta\over k+ 1},NEWLINE\]NEWLINE where the constant NEWLINE\[NEWLINE\text{Ci}\Biggl({\pi\over 2}\Biggr):= -\int^\infty_{\pi/2} {\cos t\over t} dt= 0.472\dotsNEWLINE\]NEWLINE is best possible.NEWLINENEWLINENEWLINEIt is well-known that NEWLINE\[NEWLINE\sum^\infty_{k= 1} {\cos k\theta\over k+ 1}= -1-(\cos\theta) \log\Biggl(2\sin{\theta\over 2}\Biggr)+ {\pi- \theta\over 2}\sin \theta.NEWLINE\]