The sign of the coefficients in the improved discrete weighted Hardy inequality

Abstract

For a real parameter α put bₖ(α) = C(α, k) - (-1)ᵏC((1-α)/2, k) - C((1+α)/2, k), qquad k ≥ 0. These are the coefficients of the infinite tail of lower-order terms that upgrades the discrete weighted Hardy inequality with power weight n^(α) to the improved inequality of Gupta, and the upgrade is available exactly where they are all non-negative. Gupta proved bₖ(α) ≥ 0 for α ∈ [1/3,1) ∪ {0} and conjectured a converse: for every α ∈ (0,1/3) ∪ [5,∞) some bₖ(α) is negative. Towards it he reached only the odd integers α=2m+1. We prove the conjecture in full, and more: some bₖ(α) is negative for every real α ∈ (0,1/3) and for every real α>1, so that the only real parameters at which the whole sequence stays non-negative are those of [1/3,1) ∪ {0,1}. The proof is elementary and uniform — one algebraic identity, whose requirement that a certain coefficient vanish identically is what produces the constant 1/3, together with a decay estimate for ∏ⱼ(1-(c)/(2j)) that needs nothing beyond Bernoulli's inequality. We also correct Gupta's partial result at the odd integers: its strict inequality fails at the first index it asserts, where the true value is 0, and we prove the corrected form of that strict half. All results are machine-checked in Lean 4.

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