The Quasi-Arithmetic Means

Abstract

The power means ^[r]_n (a;w), rεR, defined in the previous chapter can be looked at in the following way; for each rεR define a function φ as follows: Φ(x) = x^r, r ≠ 0, Φ(x) = log x, r = 0, then

$$M_n^{[r]}(\underline a ;\underline w ) = {\phi ^{ - 1}}\quad (\frac{1}{{{w_n}}}\sum\limits_{i = 1}^n {{w_i}\;\phi ({a_i})} ).$$

((1))

This suggests the following definitions.