A while ago I uploaded a note to the Classic Analysis section of the website on the proof of Hardy’s inequality. Hardy’s inequality says that if a function , for , then setting , we have the inequality
The discrete version of Hardy’s inequality is that if is a nonnegative sequence of reals in , for . Then
The proof for the discrete case in my note on Hardy’s inequality is incorrect as it stands–my apologies!. I will correct it later.
What I want to discuss today, though, isn’t Hardy’s inequality per se, but rather the inequality that results when you let in Hardy’s inequality. This limiting case is alternatively called the discrete Polyá-Knopp inequality or Carleman’s inequality:
If is a sequence of reals in , then
I will present two proofs of this result, which are featured in the survey “CARLEMAN’S INEQUALITY – HISTORY, PROOFS AND SOME NEW GENERALIZATIONS” by Johansson, Persson, and Wedestig. Interestingly, neither of these proofs were originally presented by Carleman. The first proof uses Hardy’s inequality and a clever limiting argument, which illustrates the claim that the Polyá-Knopp inequality is a limiting case of Hardy’s. The downside to this proof is that it doesn’t show that the inequality is strict if the sequence is not identically zero. The second is less advanced in that it just uses the strict convexity of the arithmetic mean-geometric mean (AM-GM) inequality as well as a crude estimate for , . The upside to this proof is that we get the strictness part.
Proof 1. Let . Then
Using the definition of the derivative and the continuity of the function, we see that
By the dominated convergence theorem and Hardy’s inequality,
where we use the result
Proof 2. First, an elementary lemma.
Lemma. For all , , where the inequality is strict for .
Proof. We have that for all (see my note on compound interest). Hence,
Suppose is a nonnegative sequence of reals such that converges. Then
where we use the fact that the order of summation of an absolutely convergent series is irrelevant. We can apply the AM-GM inequality to to obtain
where the last inequality follows from an application of the lemma. I claim that the inequality is in fact strict if not all the terms are zero. Indeed, suppose not. Then equality must hold in each application of the AM-GM inequality. Set . Suppose . Then
But by strict convexity, equality holds if and only if . By induction, we see that , for all . But since the harmonic series diverges, this contradicts the convergence of .