Positivity and conditional positivity of Loewner matrices

We give elementary proofs of the fact that the Loewner matrices [f(p<SUB>i</SUB>)-f(p<SUB>j</SUB>)/p<SUB>i</SUB>-p<SUB>j</SUB>]corresponding to the function f(t) = t<SUP> r</SUP> on (0, &#8734; ) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0, &#8734; ) the Loewner matrices corresponding to an operator convex function on (-1, 1) need not be conditionally negative definite