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 if with
, then
.
Proof. Let with
. Then,
has an inverse since it is strictly monotonic (since it is the composition of
and the linear function
, both of which are strictly monotonic for
). Its inverse is given by
So,