scholarly journals A Hilbert-Type Integral Inequality with Multiparameters and a Nonhomogeneous Kernel

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Qiong Liu ◽  
Wenbing Sun
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Constant Factor ◽  
Real Analysis ◽  
Type Integral ◽  
Parameter Values ◽  
Hilbert Type ◽  
Special Parameter ◽  
The Way

We first introduceΓ-function and Riemannζ-function to characterize the constant factor jointly. A Hilbert-type integral inequality with multiparameters and a nonhomogeneous kernel is given using the way of weight function and the technique of real analysis. The equivalent form is considered and its constant factors are proved to be the best possible. Some meaningful results are obtained by taking the special parameter values.

scholarly journals A Hilbert-type integral inequality with its best extension

2015 ◽  
Vol 3 (3) ◽  
pp. 121
Author(s):  
Wu Weiliang ◽  
Lian Donglan
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Constant Factor ◽  
Real Analysis ◽  
Best Value ◽  
Type Integral ◽  
Hilbert Type ◽  
The Way

<p>By using the way of weight function and the technique of real analysis, a new  Hilbert-type integral inequality with a  kernel as \(min\{x^{\lambda_1},y^{\lambda_2}\}\) and its equivalent form are established. As application, the constant factor on the plane are the best value and its best extension form with some parameters and the reverse forms are also considered.</p>

scholarly journals On a relation to two basic Hilbert-type integral inequalities

2009 ◽  
Vol 40 (3) ◽  
pp. 217-223 ◽  
Author(s):  
Bicheng Yang
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Constant Factor ◽  
Integral Inequalities ◽  
Real Analysis ◽  
Best Constant ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type

In this paper, by using the way of weight function and the technic of real analysis, a new integral inequality with some parameters and a best constant factor is given, which is a relation to two basic Hilbert-type integral inequalities. The equivalent form and the reverse forms are considered.

scholarly journals A New Hilbert-type integral inequality with a non-homogeneous kernel and its extension

2016 ◽  
Vol 4 (3) ◽  
pp. 10
Author(s):  
Weiliang Wu
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Constant Factor ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Best Value ◽  
Type Integral ◽  
Hilbert Type

By introducing some parameters , using the weight function and the technique of real analysis, a new  Hilbert-type integral inequality with a non-homogeneous kernel as \(\frac{1}{|1-axy|^{\lambda_2}}(a\geq1)\) and its equivalent form are established. As application, the constant factor on the plane is the best value and its extension form with some parameters is also considered.

scholarly journals A reverse Hilbert-type inequality with a generalized homogeneous kernel

2011 ◽  
Vol 42 (1) ◽  
pp. 1-7
Author(s):  
Bing He
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type

Inthispaper,by introducing a generalized homogeneous kernel and estimating the weight function,a new reverse Hilbert-type integral inequality with some parameters and a best constant factor is established.Furthermore, the corresponding equivalent form is considered.

scholarly journals A Hilbert-Type Integral Inequality with the Homogeneous Kernel of Degree-3 and a Best Constant Factor

2012 ◽  
Vol 4 ◽  
pp. 41-46 ◽  
Author(s):  
Zi Tian Xie ◽  
Zeng Zheng
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type

By establishing the weight function, we present a new Hilbert-type inequality with the integral in whole plane and with a best constant factor, and its kernel is a homogeneous form of degree-3, and also we put forward its equivalent form.

scholarly journals A new Hilbert-type integral inequality and the equivalent form

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Yongjin Li ◽  
Jing Wu ◽  
Bing He
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Constant Factor ◽  
Best Constant ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type

We give a new Hilbert-type integral inequality with the best constant factor by estimating the weight function. And the equivalent form is considered.

scholarly journals A reverse Hardy–Hilbert-type integral inequality involving one derivative function

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Qian Chen ◽  
Bicheng Yang
Keyword(s):  
Integral Inequality ◽  
Beta Function ◽  
Equivalent Form ◽  
Constant Factor ◽  
Weight Functions ◽  
Derivative Function ◽  
The Best Possible Constant ◽  
Type Integral ◽  
Hilbert Type ◽  
Hilbert Integral

AbstractIn this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

scholarly journals Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Dongmei Xin ◽  
Bicheng Yang ◽  
Aizhen Wang
Keyword(s):  
Integral Inequality ◽  
Beta Function ◽  
Constant Factor ◽  
Weight Functions ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Type Integral ◽  
Equivalent Property ◽  
Hilbert Type

By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

scholarly journals On a multiple Hilbert-type integral inequality involving the upper limit functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jianhua Zhong ◽  
Bicheng Yang
Keyword(s):  
Gamma Function ◽  
Integral Inequality ◽  
Constant Factor ◽  
Weight Functions ◽  
Real Analysis ◽  
Upper Limit ◽  
Type Integral ◽  
Hilbert Type

AbstractBy applying the weight functions, the idea of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.

scholarly journals On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xingshou Huang ◽  
Bicheng Yang
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant ◽  
Hilbert Type ◽  
Weight Coefficients

AbstractBy the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.

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