Discrete Copson Inequality

• Discrete Copson inequality is a family of discrete Hardy-type inequalities for sequences that employs tail-sum operators and weighted partial sums with sharp constants such as p^p for p ≥ 1.
• Different formulations use telescoping summation-by-parts, Hölder and Young's inequalities, and factorization methods to derive exact operator norms and improved remainder terms.
• Extensions include quadratic, reverse, sublinear, and negative-exponent variants, with research providing optimal weight improvements and critical remainder sequences.

Searching arXiv for papers on the discrete Copson inequality and related improvements. arXiv search query: discrete Copson inequality Das Manna improved Copson Hardy inequality The discrete Copson inequality is a family of discrete Hardy-type inequalities for sequences, centered on tail-sum operators and weighted partial sums. In the literature represented here, the term covers several closely related formulations: a finite-sum and infinite-series inequality for tail averages with sharp constant $p^p$ when $p\ge 1$ (Klaassen et al., 2020); weighted $\ell^\infty$-norm estimates for the discrete Copson operator $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$ with exact norm formula $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$ (Barza et al., 2022); and one-dimensional quadratic or $p$-power inequalities involving weighted partial sums $A_n$ and cumulative weights $Q_n$, including recent improvements in which the classical sharp constant is replaced by a strictly larger explicit weight sequence (Das et al., 1 Aug 2025). Across these formulations, the discrete Copson inequality sits in the same structural orbit as Hardy, Cesàro, and reverse Hardy inequalities, and recent work emphasizes not only optimal constants but also sharper remainder terms, critical weights, endpoint operator norms, and extensions to reverse, sublinear, and negative-exponent regimes (Das et al., 2023).

1. Classical formulations and notational conventions

A standard finite-sum form fixes $p\ge1$, nonnegative numbers $\{a_1,\dots,a_m\}$, and weights $p\ge 1$0, with

$p\ge 1$1

and

$p\ge 1$2

The discrete Copson inequality then reads

$p\ge 1$3

Letting $p\ge 1$4 yields the usual series form: if $p\ge 1$5, $p\ge 1$6, and $p\ge 1$7, then

$p\ge 1$8

This is the tail-sum formulation most directly connected with the classical Hardy–Copson duality (Klaassen et al., 2020).

A different weighted formulation, used in the one-dimensional $p\ge 1$9-power theory, takes $\ell^\infty$0, a nonnegative sequence $\ell^\infty$1, cumulative weights

$\ell^\infty$2

and partial sums

$\ell^\infty$3

In this notation, Copson’s classical inequality reads

$\ell^\infty$4

For $\ell^\infty$5, specializations of this form connect directly to discrete Hardy and power-Hardy inequalities (Das et al., 2022).

Recent work on the one-dimensional quadratic inequality uses yet another convention: $\ell^\infty$6 with parameter $\ell^\infty$7. In that setting the classical right-hand coefficient is $\ell^\infty$8, and the central problem is whether this sharp constant can nevertheless be improved by replacing it with a strictly larger sequence of weights (Das et al., 1 Aug 2025).

These notational differences are substantive rather than cosmetic. Some papers treat the Copson inequality as a tail-average estimate, some as a weighted partial-sum inequality, and some as an operator norm bound for $\ell^\infty$9. The common core is the comparison between a sequence and a weighted transform built from one-sided cumulative structure.

2. Sharp constants, duality, and proof architecture

In the series form

$C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$0

the constant $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$1 is best possible (Klaassen et al., 2020). In the weighted $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$2-power form

$C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$3

the constant $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$4 is best possible, and equality can occur only in the trivial zero case (Das et al., 2022). In the quadratic one-dimensional formulation studied by Das–Manna, the classical coefficient $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$5 is sharp in the original inequality (Das et al., 1 Aug 2025). At the endpoint $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$6, the exact operator norm of the discrete Copson operator is

$C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$7

and this formula is sharp by direct testing on tail-supported sequences (Barza et al., 2022).

The proof strategies vary with the regime. For the finite-sum and series forms, the core argument is a telescoping summation-by-parts identity augmented by Young’s inequality or Hölder’s inequality. With

$C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$8

one derives a telescoping bound for $C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}$9, then closes the estimate by Hölder or Young. The same inequality can also be derived from the fact that the discrete Copson operator is adjoint to the discrete Hardy operator, followed by $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$0–$\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$1 duality and the known sharp discrete Hardy inequality (Klaassen et al., 2020).

In the improvement theory, the proof architecture becomes more structural. One route starts from an abstract improved Hardy-type inequality

$\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$2

then specializes to the Copson setting by taking

$\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$3

The task is then to prove that the new weight $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$4 exceeds the classical Copson weight term. In practice this is done by binomial-series expansions, reduction to functions of $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$5, and convexity arguments such as showing $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$6 together with $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$7 (Das et al., 1 Aug 2025).

A second route, developed in the factorization framework, uses a generalized discrete Dirichlet Laplacian matrix $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$8. For finitely supported $\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k$9 with $p$0, one proves a quadratic form identity

$p$1

where $p$2 is the improved diagonal weight and $p$3 is a remainder matrix. Since $p$4, the diagonal term yields an improved inequality, and the factorization also supports optimality analysis (Das et al., 2023).

3. One-dimensional quadratic Copson inequalities

In the one-dimensional quadratic setting of Das–Manna, the data are a positive sequence $p$5, cumulative sums

$p$6

and weighted partial sums

$p$7

with $p$8 a complex sequence and $p$9. The classical discrete Copson inequality in this formulation carries the sharp constant $A_n$0 (Das et al., 1 Aug 2025).

The principal improvement result concerns decreasing weights. If $A_n$1 is decreasing and $A_n$2, then there is an explicit improved weight sequence $A_n$3 such that

$A_n$4

so the classical constant term is replaced by a strictly larger sequence of weights (Das et al., 1 Aug 2025).

The same paper shows that improvement is not restricted to monotone decreasing data. For the increasing choices $A_n$5 and $A_n$6, the corresponding Copson inequalities admit an improvement for

$A_n$7

respectively. In the reduced case $A_n$8, where $A_n$9 and the inequality becomes Hardy’s inequality with power weights, improvement holds for every $Q_n$0 (Das et al., 1 Aug 2025).

This corrects a natural misconception suggested by the classical sharpness statement. Sharpness of the scalar constant does not preclude a stronger inequality with a nonconstant remainder sequence. The recent theory shows that a sharp constant may coexist with a strictly larger pointwise weight on the right-hand side.

4. Improved weights and critical remainders

For $Q_n$1, Das–Manna give an explicit improved sequence. One has

$Q_n$2

where

$Q_n$3

and for $Q_n$4,

$Q_n$5

The paper states that this extends earlier work of Gupta, which covered $Q_n$6, all the way down to $Q_n$7 (Das et al., 1 Aug 2025).

In the factorization-based improvement theory, Das–Manna also isolate concrete power-weight cases. For the Copson case $Q_n$8, $Q_n$9, and for the Copson case $p\ge1$0, $p\ge1$1, they obtain explicit improved weights $p\ge1$2 and $p\ge1$3, respectively, and prove that these weights are optimal, in fact critical, in the pointwise sense: if another sequence dominates the improved weight term pointwise and the same $p\ge1$4-inequality still holds, then the two sequences must coincide identically (Das et al., 2023).

A related refinement appears in the earlier Das–Manna work on improved Hardy and Copson inequalities. In the special choice $p\ge1$5, the improved Copson inequality yields

$p\ge1$6

The same paper proves that the representation with $p\ge1$7 is best possible in the sense that no strictly larger replacement of $p\ge1$8 preserves the inequality (Das et al., 2022).

These criticality statements are stronger than mere sharpness of a scalar constant. They identify a maximal admissible pointwise remainder sequence.

5. Endpoint weighted $p\ge1$9 theory

In the endpoint theory of Barza–Demissie–Sinnamon, the discrete Copson operator is the infinite matrix $\{a_1,\dots,a_m\}$0 defined by

$\{a_1,\dots,a_m\}$1

on its natural domain

$\{a_1,\dots,a_m\}$2

For nonnegative weight sequences $\{a_1,\dots,a_m\}$3 and $\{a_1,\dots,a_m\}$4, the weighted supremum norms are

$\{a_1,\dots,a_m\}$5

The endpoint Copson inequality asks for the smallest constant $\{a_1,\dots,a_m\}$6 such that

$\{a_1,\dots,a_m\}$7

The exact answer is

$\{a_1,\dots,a_m\}$8

If one writes

$\{a_1,\dots,a_m\}$9

then $p\ge 1$00. Sharpness follows by testing on tail-supported sequences $p\ge 1$01 for $p\ge 1$02, $p\ge 1$03 otherwise (Barza et al., 2022).

For power weights

$p\ge 1$04

the paper identifies three regimes. If $p\ge 1$05, then $p\ge 1$06 and

$p\ge 1$07

If $p\ge 1$08, then $p\ge 1$09 diverges, so $p\ge 1$10 and the inequality fails. If $p\ge 1$11, then $p\ge 1$12, so $p\ge 1$13; hence $p\ge 1$14 when $p\ge 1$15, the supremum is attained at $p\ge 1$16 when $p\ge 1$17, and the knife-edge case $p\ge 1$18 gives the finite limit

$p\ge 1$19

This endpoint formulation places the discrete Copson inequality in operator-theoretic language and makes its exact norm structure completely explicit (Barza et al., 2022).

6. Sublinear, reverse, and negative-exponent variants

For $p\ge 1$20, Gao–Zhao study a reversed-direction discrete Copson inequality. With nonnegative $p\ge 1$21, cumulative sums $p\ge 1$22, and $p\ge 1$23, they prove that

$p\ge 1$24

for every nonnegative sequence $p\ge 1$25, under explicit hypotheses on the weights. One formulation is a pointwise condition; another is the global ratio bound

$p\ge 1$26

supplemented by polynomial constraints $p\ge 1$27 or $p\ge 1$28 in the parameter ranges listed in the paper. The constant $p\ge 1$29 is best possible (Gao et al., 2018).

A separate Copson-type inequality with negative exponent appears in the proof of the discrete $p\ge 1$30-Birman inequality. If $p\ge 1$31, $p\ge 1$32, $p\ge 1$33, and $p\ge 1$34, then

$p\ge 1$35

and the constant $p\ge 1$36 is best possible. The proof uses an abstract weighted $p\ge 1$37-Hardy lemma, a Gamma-function choice of auxiliary sequence, and a convexity argument for an auxiliary function $p\ge 1$38. The paper also shows that a Riemann-sum limit recovers the corresponding continuous weighted Hardy–Copson inequality, and that there is no nontrivial exact extremizer, only extremizing sequences (Štampach et al., 9 Mar 2026).

The reverse direction is also present in the broader Hardy–Copson theory. Under mild monotonicity hypotheses, Klaassen and Wellner prove a reverse Copson inequality in the probability formulation and state a reversed-Copson series estimate

$p\ge 1$39

for monotone $p\ge 1$40. They further connect the Copson operator

$p\ge 1$41

to counting-process martingales, Doob–Meyer decompositions, and the Nelson–Aalen estimator

$p\ge 1$42

with analogous backward estimators in left-censoring (Klaassen et al., 2020).

Taken together, these developments show that the discrete Copson inequality is not a single isolated estimate but a robust framework spanning sharp $p\ge 1$43 inequalities, endpoint operator norms, pointwise-improved remainders, reverse inequalities, and discrete-to-continuous limiting principles.