A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function

TL;DR

This paper introduces the concept of logarithmically absolutely monotonic functions, explores their properties, and proves the logarithmic complete monotonicity of a specific power-exponential function, expanding understanding of monotonicity classes.

Contribution

It defines logarithmically absolutely monotonic functions, proves their inclusion in absolutely monotonic functions, and establishes the logarithmic complete monotonicity of a class of power-exponential functions.

Findings

Logarithmically absolutely monotonic functions are included in absolutely monotonic functions.

The function (1 + α/x)^{x+β} is logarithmically completely monotonic under certain conditions.

A new proof shows that logarithmically completely monotonic functions are also completely monotonic.

Abstract

In the article, a notion "logarithmically absolutely monotonic function" is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function $\bigl(1+\frac{\alpha}x\bigr) ^{x+\beta}$ are proved, where $\alpha$ and $\beta$ are given real parameters, a new proof for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.