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

Can you explain why it would imply that x_k = (-1/p)^(1/p)

That’s probably a typo and it should be (-1/y)^(1/p)