---
title: Discrete Copson Inequality
url: https://www.emergentmind.com/topics/discrete-copson-inequality
type: topic
---

# Discrete Copson Inequality

Searching arXiv for recent 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 \(p\ge 1\) [2006.11818]; weighted \(\ell^\infty\)-norm estimates for the discrete Copson operator \(C^*x=(\sum_{k=n}^\infty x_k)_{n\ge1}\) with exact norm formula \(\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k\) [2211.01988]; 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 [2508.00388]. 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 [2309.04923].

## 1. Classical formulations and notational conventions

A standard finite-sum form fixes \(p\ge1\), nonnegative numbers \(\{a_1,\dots,a_m\}\), and weights \(\{p_1,\dots,p_m\}\), with
\[
P_i=\sum_{j=1}^i p_j,\qquad P_i>0,
\]
and
\[
A_n=\sum_{i=n}^m \frac{a_i p_i}{P_i},\qquad n=1,\dots,m.
\]
The discrete Copson inequality then reads
\[
\sum_{n=1}^m (A_n)^p\,p_n \le p^p \sum_{n=1}^m a_n^p\,p_n.
\]
Letting \(m\to\infty\) yields the usual series form: if \(\lambda_n\ge0\), \(\Lambda_n=\sum_{j=1}^n\lambda_j>0\), and \(a_n\ge0\), then
\[
\sum_{n=1}^\infty \biggl(\sum_{j=n}^\infty \frac{\lambda_j}{\Lambda_j}\,a_j\biggr)^p \lambda_n
\le
p^p \sum_{n=1}^\infty a_n^p \lambda_n.
\]
This is the tail-sum formulation most directly connected with the classical Hardy–Copson duality [2006.11818].

A different weighted formulation, used in the one-dimensional \(p\)-power theory, takes \(p>1\), a nonnegative sequence \(\{q_n\}\), cumulative weights
\[
Q_n=\sum_{k=1}^n q_k>0,
\]
and partial sums
\[
A_n=\sum_{k=1}^n a_k,\qquad A_0=0.
\]
In this notation, Copson’s classical inequality reads
\[
\sum_{n=1}^\infty q_n Q_n^{-c}\,|A_n|^p
\le
\Bigl(\frac{p}{p-c}\Bigr)^p
\sum_{n=1}^\infty q_n Q_n^{p-c}\,|a_n|^p,
\qquad 1<c\le p.
\]
For \(p=2\), specializations of this form connect directly to discrete Hardy and power-Hardy inequalities [2209.02612].

Recent work on the one-dimensional quadratic inequality uses yet another convention:
\[
A_n=\sum_{k=1}^n q_k a_k,\qquad Q_n=\sum_{k=1}^n q_k,\qquad A_0=0,
\]
with parameter \(\alpha\in[0,1)\). In that setting the classical right-hand coefficient is \((\alpha-1)^2/4\), and the central problem is whether this sharp constant can nevertheless be improved by replacing it with a strictly larger sequence of weights [2508.00388].

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 \(C^*\). 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
\[
\sum_{n=1}^\infty \biggl(\sum_{j=n}^\infty \frac{\lambda_j}{\Lambda_j}\,a_j\biggr)^p \lambda_n
\le
p^p \sum_{n=1}^\infty a_n^p \lambda_n,
\]
the constant \(p^p\) is best possible [2006.11818]. In the weighted \(p\)-power form
\[
\sum_{n=1}^\infty q_n Q_n^{-c}\,|A_n|^p
\le
\Bigl(\frac{p}{p-c}\Bigr)^p
\sum_{n=1}^\infty q_n Q_n^{p-c}\,|a_n|^p,
\]
the constant \(\bigl(p/(p-c)\bigr)^p\) is best possible, and equality can occur only in the trivial zero case [2209.02612]. In the quadratic one-dimensional formulation studied by Das–Manna, the classical coefficient \((\alpha-1)^2/4\) is sharp in the original inequality [2508.00388]. At the endpoint \(p=\infty\), the exact operator norm of the discrete Copson operator is
\[
C=\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k,
\]
and this formula is sharp by direct testing on tail-supported sequences [2211.01988].

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
\[
P_n=\sum_{i=1}^n p_i,\qquad
A_n=\sum_{i=n}^m a_i p_i/P_i,
\]
one derives a telescoping bound for \(\sum A_n^p p_n\), 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 \(\ell^p\)–\(\ell^q\) duality and the known sharp discrete Hardy inequality [2006.11818].

In the improvement theory, the proof architecture becomes more structural. One route starts from an abstract improved Hardy-type inequality
\[
\sum_{n=1}^\infty |A_n-A_{n-1}|^2\,\lambda_n
\ge
\sum_{n=1}^\infty w_n(\lambda,\mu)\,|A_n|^2,
\]
then specializes to the Copson setting by taking
\[
\lambda_n=q_n/Q_n^\alpha,\qquad \mu_n=Q_n^{(1-\alpha)/2}.
\]
The task is then to prove that the new weight \(w_n(\lambda,\mu)\) exceeds the classical Copson weight term. In practice this is done by binomial-series expansions, reduction to functions of \(x=1/n\), and convexity arguments such as showing \(R'''(x)>0\) together with \(R(0)=R'(0)=R''(0)=0\) [2508.00388].

A second route, developed in the factorization framework, uses a generalized discrete Dirichlet Laplacian matrix \((-\Delta_\Lambda)\). For finitely supported \(A\) with \(A_0=0\), one proves a quadratic form identity
\[
\langle A,(-\Delta_\Lambda)A\rangle_{\ell^2}
=
\langle A,EA\rangle_{\ell^2}
+
\langle RA,RA\rangle_{\ell^2},
\]
where \(E=\operatorname{diag}\{e_n\}\) is the improved diagonal weight and \(R\) is a remainder matrix. Since \(\langle RA,RA\rangle_{\ell^2}\ge0\), the diagonal term yields an improved inequality, and the factorization also supports optimality analysis [2309.04923].

## 3. One-dimensional quadratic Copson inequalities

In the one-dimensional quadratic setting of Das–Manna, the data are a positive sequence \(\{q_n\}\), cumulative sums
\[
Q_n=\sum_{k=1}^n q_k,
\]
and weighted partial sums
\[
A_n=\sum_{k=1}^n q_k a_k,\qquad A_0=0,
\]
with \(\{a_n\}\) a complex sequence and \(\alpha\in[0,1)\). The classical discrete Copson inequality in this formulation carries the sharp constant \((\alpha-1)^2/4\) [2508.00388].

The principal improvement result concerns decreasing weights. If \(\{q_n\}\) is decreasing and \(\alpha\in[1/3,1)\), then there is an explicit improved weight sequence \(w_n(q,Q)\) such that
\[
\sum_{n=1}^\infty Q_n^\alpha\,|A_n-A_{n-1}|^2\,q_n
\ge
\sum_{n=1}^\infty w_n(q,Q)\,|A_n|^2
>
\frac{(\alpha-1)^2}{4}\,(\text{classical right-hand side}),
\]
so the classical constant term is replaced by a strictly larger sequence of weights [2508.00388].

The same paper shows that improvement is not restricted to monotone decreasing data. For the increasing choices \(q_n=n\) and \(q_n=n^3\), the corresponding Copson inequalities admit an improvement for
\[
\alpha\in[17/50,1)
\quad\text{and}\quad
\alpha\in[0,1/2],
\]
respectively. In the reduced case \(q_n=1\), where \(Q_n=n\) and the inequality becomes Hardy’s inequality with power weights, improvement holds for every \(\alpha\in[0,1)\) [2508.00388].

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 \(q_n\equiv1\), Das–Manna give an explicit improved sequence. One has
\[
\sum_{n=1}^\infty n^\alpha\,|A_n-A_{n-1}|^2
\ge
\sum_{n=1}^\infty w_n(\alpha)\,|A_n|^2
>
\frac{(\alpha-1)^2}{4}\sum_{n=1}^\infty \frac{|A_n|^2}{n^{2-\alpha}},
\]
where
\[
w_1(\alpha)=1+2^\alpha-2^{\tfrac{1+\alpha}{2}},
\]
and for \(n\ge2\),
\[
w_n(\alpha)
=
n^\alpha\Bigl[1+\Bigl(1+\frac1n\Bigr)^\alpha-\Bigl(1-\frac1n\Bigr)^{\tfrac{1-\alpha}{2}}-\Bigl(1+\frac1n\Bigr)^{\tfrac{1+\alpha}{2}}\Bigr].
\]
The paper states that this extends earlier work of Gupta, which covered \(\alpha\ge1/3\), all the way down to \(\alpha\ge0\) [2508.00388].

In the factorization-based improvement theory, Das–Manna also isolate concrete power-weight cases. For the Copson case \(q_n=n^2\), \(c=3/2\), and for the Copson case \(q_n=n^3\), \(c=2\), they obtain explicit improved weights \(V_n\) and \(W_n\), 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 \(\ell^2\)-inequality still holds, then the two sequences must coincide identically [2309.04923].

A related refinement appears in the earlier Das–Manna work on improved Hardy and Copson inequalities. In the special choice \(c=3/2\), the improved Copson inequality yields
\[
\sum_{n=1}^\infty \frac{|a_n|^2}{S_n^{1/2}}
\ge
\sum_{n=1}^\infty V_n |A_n|^2
\ge
\frac1{16}\sum_{n=1}^\infty \frac{|A_n|^2}{S_n S_{n+1}},
\qquad
S_n=\frac{n(n+1)}2.
\]
The same paper proves that the representation with \(V_n\) is best possible in the sense that no strictly larger replacement of \(V_n\) preserves the inequality [2209.02612].

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

## 5. Endpoint weighted \(\ell^\infty\) theory

In the endpoint theory of Barza–Demissie–Sinnamon, the discrete Copson operator is the infinite matrix \(C^*\) defined by
\[
(C^*x)_n=\sum_{k=n}^\infty x_k,
\]
on its natural domain
\[
D(C^*)=\{x\in\ell:\forall n,\ \sum_{k=n}^\infty x_k \text{ converges in } \mathbb R\}.
\]
For nonnegative weight sequences \(u=(u_n)\) and \(v=(v_n)\), the weighted supremum norms are
\[
\|x\|_{\ell^\infty(1/u)}=\sup_{k\ge1}\frac{|x_k|}{u_k},
\qquad
\|y\|_{\ell^\infty(v)}=\sup_{n\ge1}|y_n|\,v_n.
\]
The endpoint Copson inequality asks for the smallest constant \(C\ge0\) such that
\[
\|C^*x\|_{\ell^\infty(v)}\le C\,\|x\|_{\ell^\infty(1/u)}
\qquad\text{for all }x\in D(C^*).
\]
The exact answer is
\[
C=\sup_{n\ge1} v_n\sum_{k=n}^\infty u_k.
\]
If one writes
\[
U_n=\sum_{k=n}^\infty u_k,
\]
then \(C=\|U\|_{\ell^\infty(v)}\). Sharpness follows by testing on tail-supported sequences \(x_k=u_k\) for \(k\ge m\), \(x_k=0\) otherwise [2211.01988].

For power weights
\[
u_k=k^\alpha,\qquad v_n=n^\beta,
\]
the paper identifies three regimes. If \(\alpha<-1\), then \(U_n\to0\) and
\[
C=\sum_{k=1}^\infty k^\alpha<\infty.
\]
If \(\alpha=-1\), then \(U_n=\sum_{k=n}^\infty 1/k\) diverges, so \(C=\infty\) and the inequality fails. If \(\alpha>-1\), then \(U_n\simeq n^{\alpha+1}/(\alpha+1)\), so \(v_nU_n\simeq n^{\beta+\alpha+1}/(\alpha+1)\); hence \(C=\infty\) when \(\beta+\alpha+1>0\), the supremum is attained at \(n=1\) when \(\beta+\alpha+1<0\), and the knife-edge case \(\beta+\alpha+1=0\) gives the finite limit
\[
C=\frac1{\alpha+1}.
\]
This endpoint formulation places the discrete Copson inequality in operator-theoretic language and makes its exact norm structure completely explicit [2211.01988].

## 6. Sublinear, reverse, and negative-exponent variants

For \(0<p<1\), Gao–Zhao study a reversed-direction discrete Copson inequality. With nonnegative \(\lambda_n\), cumulative sums \(\Lambda_n=\sum_{i=1}^n \lambda_i>0\), and \(L>p\), they prove that
\[
\sum_{n=1}^\infty
\Biggl(\frac1{\Lambda_n}\sum_{k=n}^\infty \lambda_k x_k\Biggr)^p
\ge
\Bigl(\frac{p}{L-p}\Bigr)^p \sum_{n=1}^\infty x_n^p
\]
for every nonnegative sequence \(\{x_n\}\), under explicit hypotheses on the weights. One formulation is a pointwise condition; another is the global ratio bound
\[
L=\sup_{n\ge1}\Bigl(\frac{\Lambda_{n+1}}{\lambda_{n+1}}-\frac{\Lambda_n}{\lambda_n}\Bigr)>p,
\]
supplemented by polynomial constraints \(a_1(L,p)\ge0\) or \(a_2(L,p)\ge0\) in the parameter ranges listed in the paper. The constant \(\bigl(p/(L-p)\bigr)^p\) is best possible [1806.07664].

A separate Copson-type inequality with negative exponent appears in the proof of the discrete \(p\)-Birman inequality. If \(1<p<\infty\), \(\alpha<0\), \(u\in C_0(\mathbb N_0)\), and \(u_0=0\), then
\[
\sum_{n=1}^\infty n^\alpha |u_n|^p
\ge
\Bigl(\frac{p-\alpha-1}{p}\Bigr)^p
\sum_{n=1}^\infty (n+1)^{\alpha-p}|u_n|^p,
\]
and the constant \(\bigl((p-\alpha-1)/p\bigr)^p\) is best possible. The proof uses an abstract weighted \(p\)-Hardy lemma, a Gamma-function choice of auxiliary sequence, and a convexity argument for an auxiliary function \(H(x)\). 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 [2603.08864].

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
\[
\sum_{n=1}^\infty
\biggl(\sum_{j=n}^\infty (\lambda_j/\Lambda_j)a_j\biggr)^p \lambda_n
\ge
\sum_{n=1}^\infty a_n^p \lambda_n
\]
for monotone \(\{a_n\}\). They further connect the Copson operator
\[
H_F^*\psi(x)=\int_{[x,\infty)} \psi(y)/F(y)\,dF(y)
\]
to counting-process martingales, Doob–Meyer decompositions, and the Nelson–Aalen estimator
\[
\widehat\Lambda_n(t)=\int_{[0,t]} (1-H_n(s-))^{-1}\,dH_n^{uc}(s),
\]
with analogous backward estimators in left-censoring [2006.11818].

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

Source: https://www.emergentmind.com/topics/discrete-copson-inequality