scholarly journals A Multiparameter Hardy–Hilbert-Type Inequality Containing Partial Sums as the Terms of Series

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jianquan Liao ◽  
Shanhe Wu ◽  
Bicheng Yang
Keyword(s):  
Type Inequality ◽  
Double Series ◽  
Constant Factor ◽  
Partial Sums ◽  
The Best Possible Constant ◽  
Hilbert Type

In this study, a multiparameter Hardy–Hilbert-type inequality for double series is established, which contains partial sums as the terms of one of the series. Based on the obtained inequality, we discuss the equivalent statements of the best possible constant factor related to several parameters. Moreover, we illustrate how the inequality obtained can generate some new Hardy–Hilbert-type inequalities.

scholarly journals On a New Half-Discrete Hilbert-Type Inequality Involving the Variable Upper Limit Integral and Partial Sums

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 229 ◽  
Author(s):  
Jianquan Liao ◽  
Shanhe Wu ◽  
Bicheng Yang
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Partial Sums ◽  
Upper Limit ◽  
The Best Possible Constant ◽  
Special Case ◽  
Equivalent Conditions ◽  
Hilbert Type

In this paper we establish a new half-discrete Hilbert-type inequality involving the variable upper limit integral and partial sums. As applications, an inequality obtained from the special case of the half-discrete Hilbert-type inequality is further investigated; moreover, the equivalent conditions of the best possible constant factor related to several parameters are proved.

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 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 A Hardy–Hilbert-Type Inequality Involving Parameters Composed of a Pair of Weight Coefficients with Their Sums

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2950
Author(s):  
Bicheng Yang ◽  
Shanhe Wu ◽  
Xingshou Huang
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
The Best Possible Constant ◽  
Equivalent Conditions ◽  
Hilbert Type ◽  
Weight Coefficients

In this paper, we establish a new Hardy–Hilbert-type inequality involving parameters composed of a pair of weight coefficients with their sum. Our result is a unified generalization of some Hardy–Hilbert-type inequalities presented in earlier papers. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed, and the equivalent forms and the operator expressions are also considered. As applications, we illustrate how the inequality obtained can generate some new Hardy–Hilbert-type inequalities.

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 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 A new discrete Hilbert-type inequality involving partial sums

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Vandanjav Adiyasuren ◽  
Tserendorj Batbold ◽  
Laith Emil Azar
Keyword(s):  
Type Inequality ◽  
Partial Sums ◽  
Hilbert Type
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