scholarly journals Equivalent Parameter Conditions for the Validity of Half-Discrete Hilbert-Type Multiple Integral Inequality with Generalized Homogeneous Kernel

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Qiang Chen ◽  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Operator Theory ◽  
Nonnegative Function ◽  
Multiple Integral ◽  
Weight Coefficient ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Equivalent Parameter ◽  
Hilbert Type ◽  
Coefficient Method

Let Gu,v be a homogeneous nonnegative function of order λ,Kn,xm,ρ=Gnλ1,xm,ρλ2. By using the weight coefficient method, the equivalent parameter conditions and best constant factors for the validity of the following half-discrete Hilbert-type multiple integral inequality ∫ℝ+m ∑n=1∞ Kn,xm,ρanfxdx≤Ma~p,αfq,β are discussed. Finally, its applications in operator theory are discussed.

The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications

2021 ◽  
Vol 19 (1) ◽  
pp. 400-411
Author(s):  
Bing He ◽  
Yong Hong ◽  
Qiang Chen
Keyword(s):  
Operator Theory ◽  
Multiple Integral ◽  
Best Constant ◽  
Equivalent Parameter ◽  
Hilbert Type

Abstract In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

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 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.

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 A Hilbert-Type Integral Inequality with the Homogeneous Kernel of Degree-3 and a Best Constant Factor

2012 ◽  
Vol 4 ◽  
pp. 41-46 ◽  
Author(s):  
Zi Tian Xie ◽  
Zeng Zheng
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type

By establishing the weight function, we present a new Hilbert-type inequality with the integral in whole plane and with a best constant factor, and its kernel is a homogeneous form of degree-3, and also we put forward its equivalent form.

scholarly journals A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

2013 ◽  
Vol 6 ◽  
pp. 38-49
Author(s):  
Zi Tian Xie ◽  
K. Raja Rama Gandhi ◽  
Zeng Zheng
Keyword(s):  
Integral Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Hilbert’S Inequality ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type ◽  
Equivalent Inequality

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

scholarly journals A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

2013 ◽  
Vol 3 ◽  
pp. 71-82
Author(s):  
Zi Tian Xie ◽  
K. Raja Rama Gandhi ◽  
Zeng Zheng
Keyword(s):  
Integral Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Hilbert’S Inequality ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type ◽  
Equivalent Inequality

In this paper, we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

scholarly journals A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

2013 ◽  
Vol 5 ◽  
pp. 49-59
Author(s):  
Zi Tian Xie ◽  
K. Raja Rama Gandhi ◽  
Zeng Zheng
Keyword(s):  
Integral Inequality ◽  
Constant Factor ◽  
Homogeneous Kernel ◽  
Best Constant ◽  
Hilbert’S Inequality ◽  
Type Integral ◽  
Best Constant Factor ◽  
Hilbert Type ◽  
Equivalent Inequality

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

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