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


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)