On improvements of the Hardy, Copson and Rellich inequalities
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910383614197760 |
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| author | Das, Bikram Manna, Atanu |
| author_facet | Das, Bikram Manna, Atanu |
| contents | Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-Δ_Λ)$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04923 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On improvements of the Hardy, Copson and Rellich inequalities Das, Bikram Manna, Atanu Functional Analysis Classical Analysis and ODEs 26D15 26D10 Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-Δ_Λ)$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained. |
| title | On improvements of the Hardy, Copson and Rellich inequalities |
| topic | Functional Analysis Classical Analysis and ODEs 26D15 26D10 |
| url | https://arxiv.org/abs/2309.04923 |