scholarly journals A Hilbert Integral-Type Inequality with Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Shang Xiaozhou ◽  
Gao Mingzhe
Keyword(s):  
Kernel Function ◽  
Integral Inequality ◽  
Type Inequality ◽  
Constant Factor ◽  
Integral Type ◽  
Type Integral ◽  
Hilbert Type ◽  
Hilbert Integral

A Hilbert-type integral inequality with parametersαand(α,λ>0)can be established by introducing a nonhomogeneous kernel function. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated. As applications, some equivalent forms are studied.

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 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 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 Equivalent Conditions of a Hilbert-Type Multiple Integral Inequality Holding

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jianquan Liao ◽  
Yong Hong ◽  
Bicheng Yang
Keyword(s):  
Integral Inequality ◽  
Type Inequality ◽  
Multiple Integral ◽  
Constant Factor ◽  
Weight Functions ◽  
Homogeneous Kernel ◽  
Optimal Constant ◽  
Equivalent Parameter ◽  
Parameter Condition ◽  
Hilbert Type

Let ∑i=1n1/pi=1pi>1, in this paper, by using the method of weight functions and technique of real analysis; it is proved that the equivalent parameter condition for the validity of multiple integral Hilbert-type inequality ∫R+nKx1,⋯,xn∏i=1nfixi dx1⋯dxn≤M∏i=1nfipi,αi with homogeneous kernel Kx1,⋯,xn of order λ is ∑i=1nαi/pi=λ+n−1, and the calculation formula of its optimal constant factor is obtained. The basic theory and method of constructing a Hilbert-type multiple integral inequality with the homogeneous kernel and optimal constant factor are solved.

scholarly journals On a Generalization of Hilbert-Type Integral Inequality

2008 ◽  
Vol 2008 ◽  
pp. 1-8
Author(s):  
Sun Baoju
Keyword(s):  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Type Integral ◽  
Hilbert Type

By introducing some parameters, we establish generalizations of the Hilbert-type inequality. As applications, the reverse and its equivalent form are considered.

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 A Hilbert-Type Integral Inequality in the Whole Plane Related to the Arc Tangent Function

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 351
Author(s):  
Michael Th. Rassias ◽  
Bicheng Yang ◽  
Andrei Raigorodskii
Keyword(s):  
Integral Inequality ◽  
Constant Factor ◽  
Cosine Function ◽  
Special Cases ◽  
Type Integral ◽  
Tangent Function ◽  
Hilbert Type

In this work we establish a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of the arc tangent function. We prove that the constant factor, which is associated with the cosine function, is optimal. Some special cases as well as some operator expressions 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.

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