Inequalities for sine sums with more variables

A result of Vietoris states that if the real numbers a1, ..., an satisfy (*) a(1) >= a(2)/2 >= . . . >= a(n)/n > 0 and a(2k-1) >= a(2k )(1 <= k <= n/2), then, for x(1), . . . x(m )> 0 with x(1)+ . . . + x(m) < pi, (**) Sigma(n)(k=1) a(k) sin(kx(1)) . . . sin(kx(m))/k(m) > 0. We prove that (**) (with ">=" instead of ">") holds under weaker conditions. It suffices to assume, instead of (*), that Sigma(N)(k=1) a(k) sin(kt)/k > 0 (N = 1, ... , n; 0< t < pi), and, moreover, (**) is valid for a larger region, namely, x(1), ..., x(m) (0, pi).