scholarly journals A More Accurate Half-Discrete Hilbert Inequality with a Nonhomogeneous Kernel

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Qiliang Huang ◽  
Bicheng Yang
Keyword(s):  
Constant Factor ◽  
Weight Functions ◽  
Hilbert Inequality ◽  
Best Constant ◽  
Best Constant Factor ◽  
The Way

By using the way of weight functions and the idea of introducing parameters, a more accurate half-discrete Hilbert inequality with a nonhomogeneous kernel and a best constant factor is presented. We also consider its best extension with the parameters, the equivalent forms, and the operator expressions as well as some reverses.

scholarly journals A New Hilbert-Type Inequality with the Homogeneous Kernel of Degree - 2 and with the Integral

2014 ◽  
Vol 7 ◽  
pp. 9-17 ◽  
Author(s):  
Zeng Zheng ◽  
Raja Rama Gandhi ◽  
Zi Tian Xie
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Weight Functions ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Hadamard’S Inequality ◽  
Best Constant Factor ◽  
Hilbert Type

By using the weight functions and by means of Hadamard's inequality, we present a new Hilbert-type inequality with the integral in whole plane, a best constant factor and a homogeneous kernel of degree-2.

scholarly journals Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Function Method ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Multiple Integral ◽  
Constant Factor ◽  
Best Constant ◽  
Necessary And Sufficient ◽  
Best Constant Factor ◽  
Hilbert Type

AbstractFor the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

Some Extensions of Hilbert-Type Inequalities

2012 ◽  
Vol 542-543 ◽  
pp. 1403-1406
Author(s):  
Bao Ju Sun
Keyword(s):  
Constant Factor ◽  
Best Constant ◽  
Best Constant Factor ◽  
Two Parameter ◽  
Hilbert Type

In this paper, an extension of Hilbert-type inequalities with a best constant factor is given by introducing two parameter .

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 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.

On a Reverse of a Hardy-Hilbert Type Inequality

2012 ◽  
Vol 170-173 ◽  
pp. 2966-2969
Author(s):  
Bao Ju Sun
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Best Constant ◽  
Two Parameters ◽  
Best Constant Factor ◽  
Hilbert Type

In this paper, a reverse of Hardy-Hilbert type inequalities with a best constant factor is given by introducing two parameters.

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