On completely monotonic functions

TL;DR

The paper investigates completely monotonic functions on $(0,\infty)$ and related Bernstein and absolutely monotonic classes, highlighting representations and interrelations. It develops a broad set of closure and construction results showing that sums, products, and compositions preserve complete monotonicity, and analyzes how exponentials and power transforms interact with these classes. A key criterion is that if $(-\log f)'$ is completely monotonic, then $f$ is completely monotonic, and the work extends to diverse composition rules such as $f\circ g$, under suitable monotonicity assumptions. The authors also provide explicit examples, including $f(x)=x^x$ on $(0,1/e)$ and $f(x)=(1/x)^{1/x}$ on $(0,\infty)$, and give a detailed parametric construction illustrating CM for a transform of the form $(x+c_0)\left(e^{ab}-\left(1+\frac{a}{x}\right)^{bx+d}\right)$ under positivity constraints.

Abstract

Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that its have completely monotonic condition.

Theorems & Definitions (55)

• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Theorem 2.9
• Corollary 2.10