scholarly journals On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1054 ◽  
Author(s):  
Bicheng Yang ◽  
Shanhe Wu ◽  
Aizhen Wang
Keyword(s):  
Summation Formula ◽  
Constant Factor ◽  
Weight Functions ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant

By means of the weight functions, the idea of introduced parameters, and the Euler-Maclaurin summation formula, a reverse half-discrete Hardy-Hilbert’s inequality and the reverse equivalent forms are given. The equivalent statements of the best possible constant factor involving several parameters are considered. As applications, two results related to the case of the non-homogeneous kernel and some particular cases are obtained.

scholarly journals Equivalent properties of a reverse half-discrete Hilbert’s inequality

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ai-zhen Wang ◽  
Bi-cheng Yang ◽  
Qiang Chen
Keyword(s):  
Summation Formula ◽  
Constant Factor ◽  
Weight Functions ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant ◽  
Equivalent Properties

Abstract By using the weight functions, the idea of introduced parameters and the Euler–Maclaurin summation formula, a reverse half-discrete Hilbert’s inequality with the homogeneous kernel and the reverse equivalent forms are given (for ${p<0}$ p < 0 , ${0< q<1}$ 0 < q < 1 ). The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, two corollaries about the case of the non-homogeneous kernel and some particular cases are obtained.

scholarly journals On a New Extended Hardy–Hilbert’s Inequality with Parameters

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 73 ◽  
Author(s):  
Bicheng Yang ◽  
Shanhe Wu ◽  
Jianquan Liao
Keyword(s):  
Summation Formula ◽  
Constant Factor ◽  
Weight Functions ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant

In this paper, by introducing parameters and weight functions, with the help of the Euler–Maclaurin summation formula, we establish the extension of Hardy–Hilbert’s inequality and its equivalent forms. The equivalent statements of the best possible constant factor related to several parameters are provided. The operator expressions and some particular cases are also discussed.

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.

scholarly journals On a Parametric Mulholland-Type Inequality and Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Bicheng Yang ◽  
Meifa Huang ◽  
Yanru Zhong
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Weight Functions ◽  
Homogeneous Kernel ◽  
The Best Possible Constant

In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are 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 A Half-Discrete Hilbert-Type Inequality in the Whole Plane with Multiparameters

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Qunwei Ma ◽  
Bicheng Yang ◽  
Leping He
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Weight Functions ◽  
Real Analysis ◽  
The Best Possible Constant ◽  
Hilbert Type

By the use of weight functions and technique of real analysis, a new half-discrete Hilbert-type inequality in the whole plane with multiparameters and the best possible constant factor is given. Furthermore, the equivalent forms, two kinds of particular inequalities, and the operator expressions with the norm are considered.

scholarly journals On two kinds of the reverse half-discrete Mulholland-type inequalities involving higher-order derivative function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiang Chen ◽  
Bicheng Yang
Keyword(s):  
Type Inequality ◽  
Higher Order ◽  
Constant Factor ◽  
Order Derivative ◽  
Weight Functions ◽  
Real Analysis ◽  
Derivative Function ◽  
Hadamard’S Inequality ◽  
The Best Possible Constant ◽  
Higher Order Derivative

AbstractBy means of the weight functions, Hermite–Hadamard’s inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also 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 reverse Mulholland-type inequality in the whole plane with general homogeneous kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ricai Luo ◽  
Bicheng Yang ◽  
Xingshou Huang
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
The Best Possible Constant ◽  
Weight Coefficients

AbstractBy using the idea of introducing parameters and weight coefficients, a new reverse discrete Mulholland-type inequality in the whole plane with general homogeneous kernel is given, which is an extension of the reverse Mulholland inequality. The equivalent forms are obtained. The equivalent statements of the best possible constant factor related to several parameters and a few applied examples are 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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