# A Continuous Description of Different Means with Application To Mixing Rules

## TL;DR

This paper introduces continuous functions to smoothly connect different types of means, useful for applications in chemistry and physics.

## Contribution

The paper proposes one-parameter functions for a continuous transition between harmonic, geometric, and arithmetic means.

## Key findings

- The Hölder mean is effective for a uniform description of mixing rules in quantum chemistry and thermophysics.
- Lehmer and Hölder means provide continuous transitions between HM, GM, and AM.
- The approach avoids discrete selection of mean types in applications.

## Abstract

Consider two positive nonzero numbers x
1 and x
2. In many cases, the
arithmetic
(AM), geometric (GM), or harmonic mean (HM) is used as an appropriate
mean x
12 of x
1 and x
2: AM = (x
1 + x
2)/2,
GM=x1x2
, and HM = GM2/AM. However, sometimes
it is not clear from the outset which mean value should be selected.
This results in a discrete problem. Instead of this, here we discuss
three one-parameter functions, which are able to describe a continuous
connection between the mean values HM, GM, and AM, respectively. Two
of these functions are known as the Lehmer mean and Hölder
mean, whereby especially the Hölder mean is suitable for generating
a uniform one-parameter description of various mixing rules as used
in quantum chemical force field calculations and thermophysical properties
calculations.