On a Sine Polynomial of Turan

In 1935, P. Turán proved that $S_{n,a}(x)= \sum_{j=1}^n{n+a-j\choose n-j} \sin(jx)>0 \quad{(n,a\in\mathbf{N}; 0<x<π).}$ 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 $x\in(0,π)$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind.

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