Properties of three functions relating to the exponential function and the existence of partitions of unity

TL;DR

This paper investigates three functions connected to the exponential function, analyzing their derivatives, monotonicity properties, and analyticity, and explores their implications for the existence of partitions of unity.

Contribution

It provides explicit derivative computations and characterizes various monotonicity properties of these functions, advancing understanding of their mathematical behavior.

Findings

Explicit derivatives of the three functions are computed.

The functions exhibit properties like analyticity and various forms of monotonicity.

Results support the existence of partitions of unity based on these functions.

Abstract

In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.