The optimal constant in Hardy-type inequalities
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite in...
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| Published in | Acta mathematica Sinica. English series Vol. 31; no. 5; pp. 731 - 754 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.05.2015
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1439-8516 1439-7617 |
| DOI | 10.1007/s10114-015-4731-5 |
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| Abstract | To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined. |
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| AbstractList | To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined. |
| Author | Chen, Mu-Fa |
| Author_xml | – sequence: 1 givenname: Mu-Fa surname: Chen fullname: Chen, Mu-Fa email: mfchen@bnu.edu.cn organization: School of Mathematical Sciences, Beijing Normal University |
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| CitedBy_id | crossref_primary_10_1155_2019_6757080 crossref_primary_10_1186_s13662_020_03006_z crossref_primary_10_1007_s10440_021_00398_2 crossref_primary_10_1186_s13662_019_2072_x crossref_primary_10_1007_s11464_018_0697_9 crossref_primary_10_1007_s10114_016_5675_0 crossref_primary_10_1090_proc_15774 |
| Cites_doi | 10.1112/jlms/s1-5.1.40 10.1007/978-3-642-18429-1 10.1112/blms/24.5.442 10.1007/BF02898239 10.1142/5513 10.1007/s11464-015-0375-0 10.1093/qmath/42.1.149 10.1007/BF02418013 10.1007/s10114-012-2316-0 10.57262/die/1367241475 10.4064/sm-44-1-31-38 10.11650/twjm/1500407479 |
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| Copyright | Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2015 |
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| DOI | 10.1007/s10114-015-4731-5 |
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| References | Chen (CR7) 2013; 29 Chen (CR6) 2005 Agapito, Paredes, Rey (CR1) 2002; 6 Muckenhoupt (CR14) 1972; XLIV Chen (CR5) 2004 Opic, Kufner (CR15) 1990 Lang, Edmunds (CR11) 2011 Manakov (CR13) 1992; 24 Chen (CR4) 2000; 43 CR12 Bennett (CR2) 1991; 42 Bliss (CR3) 1930; 5 Talenti (CR16) 1976; 110 Kufner, Maligranda, Persson (CR10) 2007 Drábek, Manśevich (CR9) 1999; 12 Chen, Wang, Zhang (CR8) 2015; 10 M F Chen (4731_CR6) 2005 V M Manakov (4731_CR13) 1992; 24 M F Chen (4731_CR5) 2004 J C C Agapito (4731_CR1) 2002; 6 G A Bliss (4731_CR3) 1930; 5 M F Chen (4731_CR8) 2015; 10 G Talenti (4731_CR16) 1976; 110 4731_CR12 G Bennett (4731_CR2) 1991; 42 B Opic (4731_CR15) 1990 A Kufner (4731_CR10) 2007 M F Chen (4731_CR7) 2013; 29 P Drábek (4731_CR9) 1999; 12 M F Chen (4731_CR4) 2000; 43 J Lang (4731_CR11) 2011 B Muckenhoupt (4731_CR14) 1972; XLIV |
| References_xml | – year: 2007 ident: CR10 publication-title: The Hardy Inequality: About its History and Some Related Results – year: 1990 ident: CR15 publication-title: Hardy-type Inequalities – volume: 5 start-page: 40 year: 1930 end-page: 46 ident: CR3 article-title: An integral inequality publication-title: J. London Math. Soc. doi: 10.1112/jlms/s1-5.1.40 – year: 2011 ident: CR11 publication-title: Eigenvalues, Embeddings and Generalised Trigonometric Functions doi: 10.1007/978-3-642-18429-1 – volume: 12 start-page: 773 year: 1999 end-page: 788 ident: CR9 article-title: On the closed solution to some nonhomogeneous eigenvalue problems with -Laplacian publication-title: Differential Integral Equations – volume: 24 start-page: 442 year: 1992 end-page: 448 ident: CR13 article-title: On the best constant in weighted inequalities for Riemann-Liouville integrals publication-title: Bull. London Math. Soc. doi: 10.1112/blms/24.5.442 – ident: CR12 – volume: 43 start-page: 1051 issue: 10 year: 2000 end-page: 1059 ident: CR4 article-title: Explicit bounds of the first eigenvalue publication-title: Sci. China, Ser. A doi: 10.1007/BF02898239 – volume: XLIV start-page: 31 year: 1972 end-page: 38 ident: CR14 article-title: Hardy’s inequality with weights publication-title: Studia Math. – year: 2004 ident: CR5 publication-title: From Markov Chains to Non-Equilibrium Particle Systems doi: 10.1142/5513 – volume: 6 start-page: 555 issue: 4 year: 2002 end-page: 563 ident: CR1 article-title: On the asymptotic behaviors of the positive solution of Δ + | | = 0 publication-title: Taiwan J. Math. – year: 2005 ident: CR6 publication-title: Eigenvalues, Inequalities, and Ergodic Theory – volume: 10 start-page: 249 issue: 2 year: 2015 end-page: 274 ident: CR8 article-title: Mixed eigenvalues of -Laplacian publication-title: Front. Math. China doi: 10.1007/s11464-015-0375-0 – volume: 42 start-page: 149 issue: 2 year: 1991 end-page: 174 ident: CR2 article-title: Some elementary inequalities III publication-title: Quart. J. Math. Oxford Ser. doi: 10.1093/qmath/42.1.149 – volume: 110 start-page: 353 issue: 1 year: 1976 end-page: 372 ident: CR16 article-title: Best constant in Sobolev inequality publication-title: Ann. Mat. Pura Appl. doi: 10.1007/BF02418013 – volume: 29 start-page: 1 issue: 1 year: 2013 end-page: 32 ident: CR7 article-title: Bilateral Hardy-type inequalities publication-title: Acta Math. Sin., Engl. Series doi: 10.1007/s10114-012-2316-0 – volume: 42 start-page: 149 issue: 2 year: 1991 ident: 4731_CR2 publication-title: Quart. J. Math. Oxford Ser. doi: 10.1093/qmath/42.1.149 – volume: 5 start-page: 40 year: 1930 ident: 4731_CR3 publication-title: J. London Math. Soc. doi: 10.1112/jlms/s1-5.1.40 – volume-title: From Markov Chains to Non-Equilibrium Particle Systems year: 2004 ident: 4731_CR5 doi: 10.1142/5513 – ident: 4731_CR12 – volume-title: The Hardy Inequality: About its History and Some Related Results year: 2007 ident: 4731_CR10 – volume: 24 start-page: 442 year: 1992 ident: 4731_CR13 publication-title: Bull. London Math. Soc. doi: 10.1112/blms/24.5.442 – volume: 43 start-page: 1051 issue: 10 year: 2000 ident: 4731_CR4 publication-title: Sci. China, Ser. A doi: 10.1007/BF02898239 – volume: 29 start-page: 1 issue: 1 year: 2013 ident: 4731_CR7 publication-title: Acta Math. Sin., Engl. Series doi: 10.1007/s10114-012-2316-0 – volume-title: Eigenvalues, Inequalities, and Ergodic Theory year: 2005 ident: 4731_CR6 – volume: 10 start-page: 249 issue: 2 year: 2015 ident: 4731_CR8 publication-title: Front. Math. China doi: 10.1007/s11464-015-0375-0 – volume: 110 start-page: 353 issue: 1 year: 1976 ident: 4731_CR16 publication-title: Ann. Mat. Pura Appl. doi: 10.1007/BF02418013 – volume-title: Hardy-type Inequalities year: 1990 ident: 4731_CR15 – volume: 12 start-page: 773 year: 1999 ident: 4731_CR9 publication-title: Differential Integral Equations doi: 10.57262/die/1367241475 – volume-title: Eigenvalues, Embeddings and Generalised Trigonometric Functions year: 2011 ident: 4731_CR11 doi: 10.1007/978-3-642-18429-1 – volume: XLIV start-page: 31 year: 1972 ident: 4731_CR14 publication-title: Studia Math. doi: 10.4064/sm-44-1-31-38 – volume: 6 start-page: 555 issue: 4 year: 2002 ident: 4731_CR1 publication-title: Taiwan J. Math. doi: 10.11650/twjm/1500407479 |
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| SubjectTerms | Approximation Constants Estimates Eulers equations Inequalities Inequality Intervals Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Optimization Studies Theorems |
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| Title | The optimal constant in Hardy-type inequalities |
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