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.