On a Sine Polynomial of Turan

In 1935, P. Tur'an proved that $$S_{n,a}(x)=\sum _{j=1}^n\left(\genfrac{}{}{0px}{}{n+a-j}{n-j}\right)\sin (jx)>0(n,a\in \mathbf{N};0<x<\pi )\mathrm{.}$$ We present various related inequalities. Among others, we show that the refinements $$S_{2n-1,a}(x)\ge \sin (x)\text{quadand}{S}_{2n,a}(x)\ge 2\sin (x)(1+\cos (x))$$ 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.