Published online by Cambridge University Press: 09 April 2009
Best-possible results are established for positivity of the partial sums of Σ sin k θ (k + α)−1. In fact odd sums are positive for −1 ≤ α ≤ α0 = 2.1 …, while 2k terms are positive on the subinterval ]0, π − 2μ0π(4k +1)−1 [, μ0 = 0.8128 …. This is analagous to the Gasper extension of the Szegö-Rogosinski-Young inequality for cosine sums.