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 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