Means in money exchange operations

TL;DR

This paper characterizes the unique weighted quasiarithmetic mean that is self reciprocally-conjugate in money exchange operations, identifying it as the weighted geometric mean.

Contribution

It proves that the only weighted quasiarithmetic mean satisfying the reciprocal-conjugate condition is the weighted geometric mean.

Findings

Weighted quasiarithmetic means must satisfy a reciprocal-conjugate condition.

The weighted geometric mean is the unique mean satisfying this condition.

The result has implications for money exchange operations.

Abstract

It is observed that in some money exchange operations, the applied $n$ -variable mean $M$ should be self reciprocally-conjugate, i.e. it should satisfy the equality \[ M\left( x_{1},\ldots,x_{n}\right) M\left( \frac{1}{x_{1}},\ldots,\frac{1}{x_{n}} \right) =1,\quad x_{1},\ldots,x_{n}>0. \] The main result says that the only weighted quasiarithmetic mean satisfying this condition is the weighet geometric mean.