An Optimal Stability Theorem for Hölder's Inequality
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913172630274048 |
|---|---|
| author | Leng, Gangsong Lu, Yifan |
| author_facet | Leng, Gangsong Lu, Yifan |
| contents | We prove an optimal $L^1$ stability theorem for Hölder's inequality. Let $p>1$, $q>1$, and $1/p+1/q=1$. If $a_k,b_k\ge 0$ and \[ \sum_{k=1}^n a_k=\sum_{k=1}^n b_k=1, \] then \[ 1-\sum_{k=1}^n a_k^{1/p}b_k^{1/q} \ge \frac1{2pq}\left(\sum_{k=1}^n |a_k-b_k|\right)^2 . \] The constant $1/(2pq)$ is best possible. We also give the corresponding integral form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31179 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Optimal Stability Theorem for Hölder's Inequality Leng, Gangsong Lu, Yifan Functional Analysis 26D15, 46E30 We prove an optimal $L^1$ stability theorem for Hölder's inequality. Let $p>1$, $q>1$, and $1/p+1/q=1$. If $a_k,b_k\ge 0$ and \[ \sum_{k=1}^n a_k=\sum_{k=1}^n b_k=1, \] then \[ 1-\sum_{k=1}^n a_k^{1/p}b_k^{1/q} \ge \frac1{2pq}\left(\sum_{k=1}^n |a_k-b_k|\right)^2 . \] The constant $1/(2pq)$ is best possible. We also give the corresponding integral form. |
| title | An Optimal Stability Theorem for Hölder's Inequality |
| topic | Functional Analysis 26D15, 46E30 |
| url | https://arxiv.org/abs/2605.31179 |