On a Sine Polynomial of Turan

Abstract

In 1935, P. Turán proved that $$S_{n,a}(x)=\sum _{j=1}^n\left(\genfrac{}{}{0ex}{}{n+a-j}{n-j}\right)\sin (jx)>0(n,a\in \mathbf{N};0<x<\pi ).$$ 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 $n\ge 1$ and real numbers $a\ge 1$ and $x\in (0,\pi )$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind.

Accepted, to appear in Rocky Mountain Journal of Mathematics