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 the inverse of we know for all in the range of , i.e., for all such that there is some such that .
By the definition of then, we have that
So, if is in the domain of then we are done. Since is the inverse of it’s domain is equal to the range of . We show that this value is in the range of using the intermediate value theorem.
Without loss of generality, assume is strictly increasing (the alternative assumption, that is strictly decreasing will produce an almost identical argument). Then, since are all positive real numbers we have . (Here if we’d assumed that was strictly decreasing the roles inequalities would be reversed.) Then we have,
Hence, by the intermediate value theorem, since
there must be some such that
Thus, is in the domain of , so