Means in money exchange operations

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.

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