Improved Vietoris Sine Inequalities for Non-Monotone, Non-Decaying Coefficients

Recently the author established an improvement of the classical Vietoris sine inequality to include sine polynomials with non-monotone coefficients. In this paper two further improvements are presented admitting sine polynomials with non-monotone and non-decaying coefficients. The extremal sums of the two results have the coefficient sequences {2a, a, 4/3, 1, 6/5, 1, 8/7, 1, ...}, where a = 0.78265..., and {3, 3/2, 7/3, 7/4, 11/5, 11/6, ...}.