Many mathematicians may never have heard of Bernstein functions, because in probability they are known as Laplace exponents, in harmonic analysis as negative definite functions, in complex analysis as Pick or Nevanlinna functions, and in matrix analysis and operator theory as

monotone function.

By (ii), we know that [xi] = [xi](x) is a

monotone function about x.

Of the calibration curves, 62-90% were best fit via the

MONOTONE function (lowest absolute SSRs), and 9-28% by MOD4P.

With LD, to get E(p) a strictly

monotone function requires a market supply function S(p) with [-S(p)] monotone.

A production function f is called quasi-sum if there exist strict

monotone functions F, [h.

Many mathematicians may not have heard of Berstein functions, because in many branches of the field, they are called different names: Laplace exponents, negative definite functions, Pick or Nevanlinna functions, operator

monotone functions, and so on.

In the article by Frank Hansen the remarkable new applications of operator

monotone functions in economics are developed, involving two notions in microeconomics, decreasing relative risk premium and risk vulnerability.

Let us remark that the set of Stieltjes transforms is a subset of the set of completely

monotone functions.

In the next section we demonstrate that a broad class of preferences introduced in [4] are represented by matrix monotone functions of order 2.

The main result in this section is that operator monotone functions are risk vulnerable.

algebras as discussed in the case of

monotone functions by [2, Theorem 5; 9, Theorem 2.