The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications
For x = ( x 1 , … , x n ) , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ , v ( y ) = ( ∑ i = 1 n b i y i ρ ) 1 / ρ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K ( u ( x )...
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| Published in | Journal of inequalities and applications Vol. 2017; no. 1; pp. 1 - 12 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
28.12.2017
Springer Nature B.V SpringerOpen |
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| Abstract | For
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, by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel
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and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered. |
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| AbstractList | Abstract For x = ( x 1 , … , x n ) ${x}= ( {x}_{1},\ldots, {x}_{{n}} )$ , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ ${u} ( {x} ) = ( \sum_{{i}=1}^{{n}} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho}$ , v ( y ) = ( ∑ i = 1 n b i y i ρ ) 1 / ρ ${v} ( {y} ) = ( \sum_{{i}=1}^{{n}} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho}$ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K ( u ( x ) , v ( y ) ) = G ( u λ 1 ( x ) v λ 2 ( y ) ) ${K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )$ and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered. For x = ( x 1 , … , x n ) , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ , v ( y ) = ( ∑ i = 1 n b i y i ρ ) 1 / ρ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K ( u ( x ) , v ( y ) ) = G ( u λ 1 ( x ) v λ 2 ( y ) ) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered. For x=(x1,…,xn), u(x)=(∑i=1naixiρ)1/ρ, v(y)=(∑i=1nbiyiρ)1/ρ, by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K(u(x),v(y))=G(uλ1(x)vλ2(y)) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered. |
| ArticleNumber | 316 |
| Author | Huang, Qiliang Hong, Yong Liao, Jianquan Yang, Bicheng |
| Author_xml | – sequence: 1 givenname: Yong surname: Hong fullname: Hong, Yong organization: School of Mathematics and Statistics, Guangdong University of Finance and Economics – sequence: 2 givenname: Qiliang surname: Huang fullname: Huang, Qiliang email: qlhuang@yeah.net organization: Department of Mathematics, Guangdong University of Education – sequence: 3 givenname: Bicheng surname: Yang fullname: Yang, Bicheng organization: Department of Mathematics, Guangdong University of Education – sequence: 4 givenname: Jianquan surname: Liao fullname: Liao, Jianquan organization: Department of Mathematics, Guangdong University of Education |
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, by using the methods and techniques of real... For x=(x1,…,xn), u(x)=(∑i=1naixiρ)1/ρ, v(y)=(∑i=1nbiyiρ)1/ρ, by using the methods and techniques of real analysis, the sufficient and necessary conditions for... Abstract For x = ( x 1 , … , x n ) ${x}= ( {x}_{1},\ldots, {x}_{{n}} )$ , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ ${u} ( {x} ) = ( \sum_{{i}=1}^{{n}} {a}_{{i}}... |
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| SubjectTerms | Analysis Applications of Mathematics best possible constant factor bounded operator Hilbert-type inequality Integrals Mathematics Mathematics and Statistics non-homogeneous kernel operator norm Operators (mathematics) sufficient and necessary conditions |
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| Title | The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications |
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