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,