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).
Show that
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