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.

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 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 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 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 Weighted hardy type integral inequalities involving many functions

2012 ◽  
Vol 43 (2) ◽  
Author(s):  
Sabir Hussain ◽  
Muhammad Amer Latif ◽  
Waseem Akhtar
Keyword(s):  
Integral Inequalities ◽  
Hardy Type ◽  
Type Integral
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