I'm having a small problem with the following lemma:
Let
a and
b be any positive numbers, and let
p and
q be any positive numbers (necessarily greater than 1) satisfying
.
Then
with equality if and only if
.
It is trivial to prove that the minimum occurs when
if we consider the function (with b fixed)
. The value of the function at the minimum is zero, and thus, this proves what comes after the statement after "then".
The problem I am currently facing is how to reason/prove (hopefully in a rigorous fashion) that the minimum is unique.
Any help is
much appreciated. Thank you.