On Positive Definiteness of Some Functions

Let [rho] be a nonnegative homogeneous function on n. General structure of the set of numerical pairs ([delta], [lambda]), for which the function (1-[rho][lambda](x))[delta]+ is positive definite on n 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(-[rho][lambda](x)) is found. We also obtain a general sufficient condition of Polya type for a function f([rho](x)) to be positive definite on n.