New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures
A non-negative function \(f:(0,\infty)\to[0,\infty)\) is called completely monotone, if it is infinitely differentiable and \((-1)^nf^{(n)}(x)\geq 0\) for all \(x>0\) and \(n\geq 1\). A non-negative function \(g:(0,\infty)\to[0,\infty)\) is a Bernstein function, if it is infinitely differentiable and \((-1)^{n-1}g^{(n)}(x)\geq 0\) for all \(x>0\) and \(n\geq 1\). A measure \(\mu\) on \([0,\infty)\) is called harmonically convex if the function \(x\mapsto x\mu[0,x]\) is convex on \((0,\infty)\). A measure \(\nu\) on \((0,\infty)\) is said to have harmonically concave tail if the function \(x\mapsto x\nu(x,\infty)\) is concave on \((0,\infty)\). The main results of the paper are the following. { Theorem 1.} For any completely monotone function \(f(x)\) and a number \(\alpha\in(0,2/3]\), there exists unique harmonically convex measure \(\mu_\alpha\) on \([0,\infty)\), such that \[ f(x)=\int_{[0,\infty)}e^{-x^{1/\alpha}t}\mu_\alpha(dt). \] { Theorem 2.} For any Bernstein function \(g(x)\) and a number \(\alpha\in(0,2/3]\), there exists unique triple \((a,b,\nu_\alpha)\), such that \[ g(x)=a+bx^{1/\alpha}+\int_{(0,\infty)}(1-e^{-x^{1/\alpha}t})\nu_\alpha(dt), \] where \(a,b\geq 0\) are constants, and \(\nu_\alpha\) is a measure on \((0,\infty)\) with harmonically concave tail. The measure \(\nu_\alpha\) satisfies the integrability condition \[ \int_{(0,\infty)}(1\wedge t)\nu_\alpha(dt)<\infty. \] Finally, the authors show that the transformations that turn a Bernstein function into one having corresponding Lévy measure with harmonically concave tail are the same as the transformations that transform a completely monotone function into one having harmonically convex measure.
scientific article; zbMATH DE number 6529751
scientific article