[For the entire collection see Zbl 0760.00004.]
The tail probabilities of two weighted sums of independent gamma random variables are compared when the first vector of weights majorizes the second vector of weights. The conjecture that the two cumulative distribution functions cross exactly once is established in four special cases by means of the variation-diminishing property of totally positive kernels. Bounds are obtained for the location of the unique crossing point and its asymptotic behavior is determined.