Improved Discrete Dual p-Hardy and Weighted Discrete p- Birman Inequalities

Abstract

In this paper, we establish a new version of one dimensional generalized discrete dual p-Hardy inequality with a shift. Using this generalized discrete dual p-Hardy inequality, we obtain improvements of two discrete dual p-Hardy inequalities. To be specific, for p>1 and A∈ C_c(ℕ₀) satisfying A₀=A₁=0, we first improve the discrete dual p-Hardy inequality &∑_n=2^∞(n−1)^p| A_n−A_n−1|^p≥1/p^p∑_n=2^∞|A_n|^p, where the associate constant term is sharp. Subsequently, we improve its power-type weighted discrete dual p-Hardy extension &∑_n=2^∞(n−1)^α|A_n−A_n−1|^p≥((α+1−p)/p)^p ∑_n=2^∞|A_n|^p/(n^p−α) for p−1<α≤ p, where the associated constant term is also sharp. We also establish a discrete p- Birman inequality with power weights. Furthermore, we establish a multivariable dual p-Hardy inequality with a sharp constant. The proof proceeds by first establishing the inequality for two variables and then extending the argument to multiple variables, while preserving the sharpness of the constant.

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BibTeX RIS

BibTeX

@misc{das2026improved,
  title = {{Improved Discrete Dual $p$-Hardy and Weighted Discrete $p$- Birman Inequalities}},
  author = {Bikram Das},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11890},
  eprint = {2609.11890},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11890},
}

RIS

TY  - UNPB
TI  - Improved Discrete Dual p-Hardy and Weighted Discrete p- Birman Inequalities
AU  - Bikram Das
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11890
UR  - https://arxiv.org/abs/2609.11890
AB  - In this paper, we establish a new version of one dimensional generalized discrete dual p-Hardy inequality with a shift. Using this generalized discrete dual p-Hardy inequality, we obtain improvements of two discrete dual p-Hardy inequalities. To be specific, for p>1 and A∈ C_c(ℕ_0) satisfying A_0=A_1=0, we first improve the discrete dual p-Hardy inequality &∑_n=2^∞(n−1)^p| A_n−A_n−1|^p≥1/p^p∑_n=2^∞|A_n|^p, where the associate constant term is sharp. Subsequently, we improve its power-type weighted discrete dual p-Hardy extension &∑_n=2^∞(n−1)^α|A_n−A_n−1|^p≥((α+1−p)/p)^p ∑_n=2^∞|A_n|^p/(n^p−α) for p−1<α≤ p, where the associated constant term is also sharp. We also establish a discrete p- Birman inequality with power weights. Furthermore, we establish a multivariable dual p-Hardy inequality with a sharp constant. The proof proceeds by first establishing the inequality for two variables and then extending the argument to multiple variables, while preserving the sharpness of the constant.
ER  -

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