  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, ...}.
