On Positive Definiteness of Some Functions

Let ρ be a nonnegative homogeneous function on Rn. General structure of the set of numerical pairs (δ, λ), for which the function (1−ρλ(x))δ+ is positive definite on Rn is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic functions; a connection of this problem with the Schoenberg problem on positive definiteness of the function exp(−ρλ(x)) is found. We also obtain a general sufficient condition of Polya type for a function f(ρ(x)) to be positive definite on Rn.