On a sine polynomial of Turán

Abstract

In 1935, Tur\'an proved that \[ S_{n,a}(x)= \sum _{j=1}^n{n+a-j\choose n-j} \sin (jx)>0, \] \[n,a\in \mathbf {N},\quad 0\lt x\lt \pi .\] We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin (x) \quad \mbox {and} \quad {S_{2n,a}(x)\geq 2\sin (x)(1+\cos (x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in (0,\pi )$. Moreover, we apply our theorems on sine sums to obtain inequalities for Chebyshev polynomials of the second kind.