A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications
Let x = ( x 1 , x 2 , … , x n ) , and let K ( u ( x ) , v ( y ) ) satisfy u ( r x ) = r u ( x ) , v ( r y ) = r v ( y ) , K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) , and K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) . In this paper, we obtain a necessary and sufficient condition and the best...
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| Published in | Journal of inequalities and applications Vol. 2020; no. 1; pp. 1 - 13 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
13.05.2020
Springer Nature B.V SpringerOpen |
| Subjects | |
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| Abstract | Let
x
=
(
x
1
,
x
2
,
…
,
x
n
)
, and let
K
(
u
(
x
)
,
v
(
y
)
)
satisfy
u
(
r
x
)
=
r
u
(
x
)
,
v
(
r
y
)
=
r
v
(
y
)
,
K
(
r
u
,
v
)
=
r
λ
λ
1
K
(
u
,
r
−
λ
1
λ
2
v
)
, and
K
(
u
,
r
v
)
=
r
λ
λ
2
K
(
r
−
λ
2
λ
1
u
,
v
)
. In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel
K
(
u
(
x
)
,
v
(
y
)
)
and discuss its applications in the theory of operators. |
|---|---|
| AbstractList | Abstract Let x = ( x 1 , x 2 , … , x n ) $x=(x_{1},x_{2},\ldots,x_{n})$ , and let K ( u ( x ) , v ( y ) ) $K(u(x),v(y))$ satisfy u ( r x ) = r u ( x ) $u(rx)=ru(x)$ , v ( r y ) = r v ( y ) $v(ry)=rv(y)$ , K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$ , and K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$ . In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel K ( u ( x ) , v ( y ) ) $K(u(x),v(y))$ and discuss its applications in the theory of operators. Abstract Let $x=(x_{1},x_{2},\ldots,x_{n})$ x = ( x 1 , x 2 , … , x n ) , and let $K(u(x),v(y))$ K ( u ( x ) , v ( y ) ) satisfy $u(rx)=ru(x)$ u ( r x ) = r u ( x ) , $v(ry)=rv(y)$ v ( r y ) = r v ( y ) , $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$ K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) , and $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$ K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) . In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel $K(u(x),v(y))$ K ( u ( x ) , v ( y ) ) and discuss its applications in the theory of operators. Let x = ( x 1 , x 2 , … , x n ) , and let K ( u ( x ) , v ( y ) ) satisfy u ( r x ) = r u ( x ) , v ( r y ) = r v ( y ) , K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) , and K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) . In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel K ( u ( x ) , v ( y ) ) and discuss its applications in the theory of operators. Let x=(x1,x2,…,xn), and let K(u(x),v(y)) satisfy u(rx)=ru(x), v(ry)=rv(y), K(ru,v)=rλλ1K(u,r−λ1λ2v), and K(u,rv)=rλλ2K(r−λ2λ1u,v). In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel K(u(x),v(y)) and discuss its applications in the theory of operators. |
| ArticleNumber | 140 |
| Author | Hong, Yong Chen, Qiang Liao, Jianquan Yang, Bicheng |
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| Cites_doi | 10.1186/s13660-016-1020-5 10.1155/2009/192197 10.1155/2007/27962 10.1186/s13660-015-0861-7 10.1186/s13660-016-1026-z 10.1186/s13660-016-1075-3 |
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| Keywords | Necessary and sufficient condition Generalized homogeneous kernel Hilbert-type multiple integral inequality Operator norm Bounded operator The best constant factor 26D15 |
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| References | Hong (CR15) 2006; 2006 Zhong, Yang (CR10) 2007; 2007 Xin, Yang, Chen (CR11) 2016; 2016 Fichtigoloz (CR17) 1957 Yang (CR4) 2004; 1 Hong (CR1) 2014; 57 Perić, Vuković (CR3) 2009; 12 Yang, Chen (CR14) 2016; 2016 Hong, Wen (CR5) 2017; 37A Kuang (CR12) 2004 Huang, Yang (CR16) 2009; 2009 Yang, Qiang (CR8) 2016; 2016 Rassias, Yang (CR6) 2016; 7 Yang, Chen (CR13) 2015; 2015 Chen, Shi, Yang (CR7) 2016; 2016 Yang (CR9) 2015; 18 He, Cao, Yang (CR2) 2015; 58 G.H. Fichtigoloz (2401_CR17) 1957 B.C. Yang (2401_CR9) 2015; 18 Q.L. Huang (2401_CR16) 2009; 2009 Y. Hong (2401_CR1) 2014; 57 B.C. Yang (2401_CR13) 2015; 2015 Q. Chen (2401_CR7) 2016; 2016 I. Perić (2401_CR3) 2009; 12 B.C. Yang (2401_CR4) 2004; 1 M.Th. Rassias (2401_CR6) 2016; 7 J.C. Kuang (2401_CR12) 2004 Y. Hong (2401_CR5) 2017; 37A D.M. Xin (2401_CR11) 2016; 2016 B.C. Yang (2401_CR14) 2016; 2016 B. He (2401_CR2) 2015; 58 Y. Hong (2401_CR15) 2006; 2006 B.C. Yang (2401_CR8) 2016; 2016 W.Y. Zhong (2401_CR10) 2007; 2007 |
| References_xml | – volume: 58 start-page: 661 year: 2015 end-page: 672 ident: CR2 article-title: A brand new multiple Hilbert-type integral inequality publication-title: Acta Math. Sin. Chin. Ser. contributor: fullname: Yang – volume: 1 start-page: 1 issue: 1 year: 2004 end-page: 8 ident: CR4 article-title: On an extension of Hilbert’s integral inequality with some parameters publication-title: Aust. J. Math. Anal. Appl. contributor: fullname: Yang – volume: 2016 year: 2016 ident: CR7 article-title: A relation between two simple Hardy–Mulholland-type inequalities with parameters publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1020-5 contributor: fullname: Yang – volume: 57 start-page: 833 year: 2014 end-page: 840 ident: CR1 article-title: A Hilbert-type integral inequality with quasi-homogeneous kernel and several functions publication-title: Acta Math. Sin. Chin. Ser. contributor: fullname: Hong – volume: 37A start-page: 329 issue: 3 year: 2017 end-page: 336 ident: CR5 article-title: A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel and the best constant factor publication-title: Chin. Ann. Math. contributor: fullname: Wen – volume: 2009 year: 2009 ident: CR16 article-title: On a multiple Hilbert-type integral operator and applications publication-title: J. Inequal. Appl. doi: 10.1155/2009/192197 contributor: fullname: Yang – volume: 2016 year: 2016 ident: CR14 article-title: A new extension of Hardy–Hilbert’s inequality in the whole plane publication-title: J. Funct. Spaces contributor: fullname: Chen – volume: 18 start-page: 429 year: 2015 end-page: 441 ident: CR9 article-title: On a more accurate multidimensional Hilbert-type inequality with parameters publication-title: Math. Inequal. Appl. contributor: fullname: Yang – volume: 2007 year: 2007 ident: CR10 article-title: On multiple Hardy–Hilbert’s integral inequality with kernel publication-title: J. Inequal. Appl. doi: 10.1155/2007/27962 contributor: fullname: Yang – volume: 2015 year: 2015 ident: CR13 article-title: On a Hardy–Hilbert-type inequality with parameters publication-title: J. Inequal. Appl. doi: 10.1186/s13660-015-0861-7 contributor: fullname: Chen – volume: 2006 year: 2006 ident: CR15 article-title: On multiple Hardy–Hilbert integral inequalities with some parameters publication-title: J. Inequal. Appl. contributor: fullname: Hong – year: 1957 ident: CR17 publication-title: A Course in Differential and Integral Calculus contributor: fullname: Fichtigoloz – volume: 7 start-page: 249 issue: 1 year: 2016 end-page: 269 ident: CR6 article-title: On a Hardy–Hilbert-type inequality with a general homogeneous kernel publication-title: Int. J. Nonlinear Anal. Appl. contributor: fullname: Yang – year: 2004 ident: CR12 publication-title: Applied Inequalities contributor: fullname: Kuang – volume: 2016 year: 2016 ident: CR8 article-title: On a more accurate Hardy–Mulholland-type inequality publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1026-z contributor: fullname: Qiang – volume: 2016 year: 2016 ident: CR11 article-title: A discrete Hilbert-type inequality in the whole plane publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1075-3 contributor: fullname: Chen – volume: 12 start-page: 525 year: 2009 end-page: 536 ident: CR3 article-title: Hardy–Hilbert’s inequalities with a general homogeneous kernel publication-title: Math. Inequal. Appl. contributor: fullname: Vuković – volume: 18 start-page: 429 year: 2015 ident: 2401_CR9 publication-title: Math. Inequal. Appl. contributor: fullname: B.C. Yang – volume: 2015 year: 2015 ident: 2401_CR13 publication-title: J. Inequal. Appl. doi: 10.1186/s13660-015-0861-7 contributor: fullname: B.C. Yang – volume: 7 start-page: 249 issue: 1 year: 2016 ident: 2401_CR6 publication-title: Int. J. Nonlinear Anal. Appl. contributor: fullname: M.Th. Rassias – volume: 1 start-page: 1 issue: 1 year: 2004 ident: 2401_CR4 publication-title: Aust. J. Math. Anal. Appl. contributor: fullname: B.C. Yang – volume: 58 start-page: 661 year: 2015 ident: 2401_CR2 publication-title: Acta Math. Sin. Chin. Ser. contributor: fullname: B. He – volume: 2016 year: 2016 ident: 2401_CR14 publication-title: J. Funct. Spaces contributor: fullname: B.C. Yang – volume-title: A Course in Differential and Integral Calculus year: 1957 ident: 2401_CR17 contributor: fullname: G.H. Fichtigoloz – volume: 2016 year: 2016 ident: 2401_CR8 publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1026-z contributor: fullname: B.C. Yang – volume: 2006 year: 2006 ident: 2401_CR15 publication-title: J. Inequal. Appl. contributor: fullname: Y. Hong – volume-title: Applied Inequalities year: 2004 ident: 2401_CR12 contributor: fullname: J.C. Kuang – volume: 2009 year: 2009 ident: 2401_CR16 publication-title: J. Inequal. Appl. doi: 10.1155/2009/192197 contributor: fullname: Q.L. Huang – volume: 37A start-page: 329 issue: 3 year: 2017 ident: 2401_CR5 publication-title: Chin. Ann. Math. contributor: fullname: Y. Hong – volume: 12 start-page: 525 year: 2009 ident: 2401_CR3 publication-title: Math. Inequal. Appl. contributor: fullname: I. Perić – volume: 2016 year: 2016 ident: 2401_CR11 publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1075-3 contributor: fullname: D.M. Xin – volume: 2007 year: 2007 ident: 2401_CR10 publication-title: J. Inequal. Appl. doi: 10.1155/2007/27962 contributor: fullname: W.Y. Zhong – volume: 57 start-page: 833 year: 2014 ident: 2401_CR1 publication-title: Acta Math. Sin. Chin. Ser. contributor: fullname: Y. Hong – volume: 2016 year: 2016 ident: 2401_CR7 publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1020-5 contributor: fullname: Q. Chen |
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| Score | 2.30075 |
| Snippet | Let
x
=
(
x
1
,
x
2
,
…
,
x
n
)
, and let
K
(
u
(
x
)
,
v
(
y
)
)
satisfy
u
(
r
x
)
=
r
u
(
x
)
,
v
(
r
y
)
=
r
v
(
y
)
,
K
(
r
u
,
v
)
=
r
λ
λ
1
K
(
u
,
r
−
λ... Abstract Let $x=(x_{1},x_{2},\ldots,x_{n})$ x = ( x 1 , x 2 , … , x n ) , and let $K(u(x),v(y))$ K ( u ( x ) , v ( y ) ) satisfy $u(rx)=ru(x)$ u ( r x ) = r u... Let x=(x1,x2,…,xn), and let K(u(x),v(y)) satisfy u(rx)=ru(x), v(ry)=rv(y), K(ru,v)=rλλ1K(u,r−λ1λ2v), and K(u,rv)=rλλ2K(r−λ2λ1u,v). In this paper, we obtain a... Abstract Let x = ( x 1 , x 2 , … , x n ) $x=(x_{1},x_{2},\ldots,x_{n})$ , and let K ( u ( x ) , v ( y ) ) $K(u(x),v(y))$ satisfy u ( r x ) = r u ( x )... |
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| SubjectTerms | Analysis Applications of Mathematics Bounded operator Generalized homogeneous kernel Hilbert-type multiple integral inequality Integrals Kernels Mathematics Mathematics and Statistics Necessary and sufficient condition Operator norm Operators (mathematics) The best constant factor |
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| Title | A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications |
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