An improved Vietoris sine inequality

We prove that if {ak}{ak} is a sequence of positive numbers, such that (2j−1)j+12jja2j+1≤a2j≤2j−12ja2j−1 for all j=1,2,…j=1,2,…, then for all n=1,2,…n=1,2,…, x∈[0,π]x∈[0,π], ∑k=1naksin(kx)≥0. An example is {ak}={1,12,12,342,13,563,…}={1,0.5,0.707,0.530,0.577,0.481,…} where ak=1/(k+1)/2 for odd kk, and (k−1)/(kk/2), for even kk. This improves the well-known Vietoris sine inequality, by relaxing the requirement that {an}{an} has to be a nonincreasing sequence.The proof is based on a Lukács-type inequality and a result on positive trigonometric sums with “convex” coefficients (both established recently by the authors), the classical Sturm Theorem on the number of real roots of a polynomial, and a well-known comparison principle. The symbolic manipulation software MAPLE is used amply for various computations.