Let be a continuous, strictly monotonic function on with inverse , and let be given positive real numbers. Then define,
This is called the mean of with respect to . (When for , this coincides with the th power mean from this exercise).
Proof. Since is strictly monotonic on the positive real axis and are positive reals, we know is strictly increasing or strictly decreasing, and correspondingly we have,
First, assume is striclty increasing, then
Since is strictly increasing so is its inverse (by Apostol’s Theorem 3.10); thus, we have
If is strictly decreasing then