On Copson’s inequalities for 0 < p < 1 $0< p<1$
Abstract Let ( λ n ) n ≥ 1 $(\lambda_{n})_{n \geq1}$ be a positive sequence and let Λ n = ∑ i = 1 n λ i $\varLambda_{n}=\sum^{n}_{i=1}\lambda_{i}$ . We study the following Copson inequality for 0 < p < 1 $0< p<1$ , L > p $L>p$ : ∑ n = 1 ∞ ( 1 Λ n ∑ k = n ∞ λ k x k ) p ≥ ( p L − p )...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-03-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-020-02339-3 |
| _version_ | 1818036816419225600 |
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| author | Peng Gao HuaYu Zhao |
| author_facet | Peng Gao HuaYu Zhao |
| author_sort | Peng Gao |
| collection | DOAJ |
| description | Abstract Let ( λ n ) n ≥ 1 $(\lambda_{n})_{n \geq1}$ be a positive sequence and let Λ n = ∑ i = 1 n λ i $\varLambda_{n}=\sum^{n}_{i=1}\lambda_{i}$ . We study the following Copson inequality for 0 < p < 1 $0< p<1$ , L > p $L>p$ : ∑ n = 1 ∞ ( 1 Λ n ∑ k = n ∞ λ k x k ) p ≥ ( p L − p ) p ∑ n = 1 ∞ x n p . $$\begin{aligned} \sum^{\infty}_{n=1} \Biggl(\frac{1}{\varLambda_{n}} \sum^{\infty }_{k=n}\lambda_{k} x_{k} \Biggr)^{p} \geq \biggl( \frac{p}{L-p} \biggr)^{p} \sum^{\infty}_{n=1}x^{p}_{n}. \end{aligned}$$ We find conditions on λ n $\lambda_{n}$ such that the above inequality is valid with the constant being the best possible. |
| first_indexed | 2024-12-10T07:16:58Z |
| format | Article |
| id | doaj.art-a1bcc13d38024ef8ae7e339a8fa71125 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-10T07:16:58Z |
| publishDate | 2020-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-a1bcc13d38024ef8ae7e339a8fa711252022-12-22T01:57:55ZengSpringerOpenJournal of Inequalities and Applications1029-242X2020-03-012020111310.1186/s13660-020-02339-3On Copson’s inequalities for 0 < p < 1 $0< p<1$Peng Gao0HuaYu Zhao1Department of Mathematics, School of Mathematical Sciences, Beihang UniversityAcademy of Mathematics and Systems Science, Chinese Academy of SciencesAbstract Let ( λ n ) n ≥ 1 $(\lambda_{n})_{n \geq1}$ be a positive sequence and let Λ n = ∑ i = 1 n λ i $\varLambda_{n}=\sum^{n}_{i=1}\lambda_{i}$ . We study the following Copson inequality for 0 < p < 1 $0< p<1$ , L > p $L>p$ : ∑ n = 1 ∞ ( 1 Λ n ∑ k = n ∞ λ k x k ) p ≥ ( p L − p ) p ∑ n = 1 ∞ x n p . $$\begin{aligned} \sum^{\infty}_{n=1} \Biggl(\frac{1}{\varLambda_{n}} \sum^{\infty }_{k=n}\lambda_{k} x_{k} \Biggr)^{p} \geq \biggl( \frac{p}{L-p} \biggr)^{p} \sum^{\infty}_{n=1}x^{p}_{n}. \end{aligned}$$ We find conditions on λ n $\lambda_{n}$ such that the above inequality is valid with the constant being the best possible.http://link.springer.com/article/10.1186/s13660-020-02339-3Copson’s inequalities |
| spellingShingle | Peng Gao HuaYu Zhao On Copson’s inequalities for 0 < p < 1 $0< p<1$ Journal of Inequalities and Applications Copson’s inequalities |
| title | On Copson’s inequalities for 0 < p < 1 $0< p<1$ |
| title_full | On Copson’s inequalities for 0 < p < 1 $0< p<1$ |
| title_fullStr | On Copson’s inequalities for 0 < p < 1 $0< p<1$ |
| title_full_unstemmed | On Copson’s inequalities for 0 < p < 1 $0< p<1$ |
| title_short | On Copson’s inequalities for 0 < p < 1 $0< p<1$ |
| title_sort | on copson s inequalities for 0 p 1 0 p 1 |
| topic | Copson’s inequalities |
| url | http://link.springer.com/article/10.1186/s13660-020-02339-3 |
| work_keys_str_mv | AT penggao oncopsonsinequalitiesfor0p10p1 AT huayuzhao oncopsonsinequalitiesfor0p10p1 |