Barycenters of extreme points in the cone of non-negative entire functions (Q1084227)

Consider the cone K of all entire functions whose restriction to \({\mathbb{R}}\) is nonnegative. A function f is an internal point of K when f has no real zeros, f is a boundary point of K when f has at least one real zero and f is an extreme point of K when all the zeros of f are real. A function f(z)\(\in K\) is called the barycenter of a finite number of extreme linearly independent functions \(H_ 1(z),...,H_ n(z)\) of K if there is a relation of the form \[ (*)\quad f(z)+H_ 1(z)+...+H_ n(z)=0. \] In the present note, the author proves that the function \[ f(z)=2e+\exp (e^{\sqrt{z}})+\exp (e^{-\sqrt{z}}) \] is not the barycenter of extreme points of K. First of all, it is shown that f(z) is an entire function and that f(x)\(\geq 0\) for real x. Moreover, if f(z) satisfies the relation (*) for extreme points \(H_ 1,...,H_ n\) of K, then by application of a simplified form of a lemma of \textit{G. Hiromi} and \textit{M. Ozawa} [Kōdai Math. Sem. Reports 17, 281-306 (1965; Zbl 0154.079)], the author proves that the functions \(H_ 1(z),...,H_ n(z)\) are linearly dependent.