We recall the definition of the th power mean .
For , and with , we define the th power-mean as:
Now, for , prove for not all equal.
Proof. From the Cauchy-Schwarz inequality we know that for real numbers and , we have
with equality if and only if there is some such that for all . Letting and we have
This inequality is strict since if equality held there would exist some such that for all , but this would imply for all , contradicting our assumption that the are not all equal. Since (see here), this implies