scholarly journals A Hilbert's Inequality with a Best Constant Factor

2009 ◽  
Vol 2009 (1) ◽  
pp. 820176 ◽  
Author(s):  
Zheng Zeng ◽  
Zi-tian Xie
Keyword(s):  
Constant Factor ◽  
Best Constant ◽  
Hilbert’S Inequality ◽  
Best Constant Factor

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.

scholarly journals On inequalities of Hilbert's type

2007 ◽  
Vol 76 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Yongjin Li ◽  
Bing He
Keyword(s):  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Best Constant ◽  
Hilbert’S Inequality ◽  
Type Integral ◽  
Best Constant Factor ◽  
Special Case ◽  
New Inequalities

By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

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 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.

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