An extension of the Fejer-Jackson inequality

Best-possible results are established for positivity of the partial sums of Sigma sink theta(k + alpha)(-1). In fact odd sums are positive for -1 less than or equal to alpha less than or equal to alpha(0) = 2.1..., while sums with 2k terms are positive on the subinterval]0, pi - 2 mu(0) pi(4k + 1)(-1) [, mu(0) = 0.8128....This is analagous to the Gasper extension of the Szego-Rogosinski-Young inequality for cosine sums.