scholarly journals Parameterized Hilbert-Type Integral Inequalities in the Whole Plane

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Qiliang Huang ◽  
Shanhe Wu ◽  
Bicheng Yang
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Integral Inequalities ◽  
Weight Functions ◽  
Real Analysis ◽  
Type Integral ◽  
Hilbert Type ◽  
The Way

By the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters. The constant factors related to the hypergeometric function and the beta function are proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases in the homogeneous kernels.

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 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 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 Hermite-Hadamard Type Integral Inequalities for Functions Whose Second-Order Mixed Derivatives Are Coordinated(s,m)-P-Convex

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yu-Mei Bai ◽  
Shan-He Wu ◽  
Ying Wu
Keyword(s):  
Hypergeometric Function ◽  
Convex Functions ◽  
Beta Function ◽  
Second Order ◽  
Integral Inequalities ◽  
Type Integral ◽  
Mixed Derivatives ◽  
Expression Form

We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated(s,m)-P-convex. An expression form of Hermite-Hadamard type integral inequalities via the beta function and the hypergeometric function is also presented. Our results provide a significant complement to the work of Wu et al. involving the Hermite-Hadamard type inequalities for coordinated(s,m)-P-convex functions in an earlier article.

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 Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta function

2018 ◽  
Vol 12 (2) ◽  
pp. 273-296 ◽  
Author(s):  
Michael Rassias ◽  
Bicheng Yang ◽  
Andrei Raigorodskii
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integral Inequalities ◽  
Weight Functions ◽  
Real Analysis ◽  
Hardy Type ◽  
Type Integral ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Equivalent Conditions

Applying techniques of real analysis and weight functions, we study some equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular nonhomogeneous kernel. The constant factors are related to the Riemann zeta function and are proved to be best possible. In the form of applications, we deduce a few equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular homogeneous kernel. We also consider some corollaries as particular cases.

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 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 Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1006
Author(s):  
Michael Th. Rassias ◽  
Bicheng Yang ◽  
Andrei Raigorodskii
Keyword(s):  
Integral Inequalities ◽  
Weight Functions ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Hardy Type ◽  
Type Integral ◽  
Equivalent Properties ◽  
Equivalent Conditions

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

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