  Given positive numbers p_1 < p_2 < ... < p_n, and a real number r let L_r be
the n by n matrix with its (i,j) entry equal to (p_i^r-p_j^r)/(p_i-p_j). A
well-known theorem of C. Loewner says that L_r is positive definite when 0 < r
< 1. In contrast, R. Bhatia and J. Holbrook, (Indiana Univ. Math. J, 49 (2000)
1153-1173) showed that when 1 < r < 2, the matrix L_r has only one positive
eigenvalue, and made a conjecture about the signatures of eigenvalues of L_r
for other r. That conjecture is proved in this paper.
