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