An improved Vietoris sine inequality (Q476125)

This is a nice paper that presents a relaxation of the classical Vietoris sine inequality. The classical inequality states that if \(a_k\) is a non-increasing sequence, converging to 0, with \(a_{2j} \leq \frac{2j-1}{2j}a_{2j-1}\) for all \(j \in \mathbb{N}\), then the partial sums of the series \(\sum a_k \sin(kx)\) are non-negative for all \(x \in [0,\pi]\). \vskip .1 in In this paper, the author proves that the partial sums of the series \(\sum a_k \sin(kx)\), \(x \in [0,\pi]\) are non-negative if \[ \frac{(2j-1)\sqrt{j+1}}{2j\sqrt{j}}a_{2j+1} \leq a_{2j} \leq \frac{2j-1}{2j}a_{2j-1},\qquad j \in \mathbb{N}. \] Thus, the condition that \(a_k\) converges monotonically to \(0\) in the classical inequality is relaxed in this new result. \vskip .1 in The proof of the result uses Sturm's theorem, along with Maple software, to prove that specific polynomials are non-negative. Several examples of sequences \(a_k\) satisfying the above conditions are given.