Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey

2014 ◽  
pp. 17-52
Author(s):  
Vandanjav Adiyasuren ◽  
Tserendorj Batbold ◽  
Mario Krnić
Keyword(s):  
Hilbert Type

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 New dynamic Hilbert-type inequalities in two independent variables involving Fenchel–Legendre transform

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. A. El-Deeb ◽  
Saima Rashid ◽  
Zareen A. Khan ◽  
S. D. Makharesh
Keyword(s):  
Time Scales ◽  
Legendre Transform ◽  
Independent Variables ◽  
Hilbert Type ◽  
Two Independent Variables ◽  
New Inequalities

AbstractIn this paper, we establish some dynamic Hilbert-type inequalities in two independent variables on time scales by using the Fenchel–Legendre transform. We also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases. Our results give more general forms of several previously established inequalities.

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 Hilbert-type inequalities for time scale nabla calculus

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
H. M. Rezk ◽  
Ghada AlNemer ◽  
H. A. Abd El-Hamid ◽  
Abdel-Haleem Abdel-Aty ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Arbitrary Time ◽  
Hilbert Type

Abstract This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

scholarly journals On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xingshou Huang ◽  
Bicheng Yang
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant ◽  
Hilbert Type ◽  
Weight Coefficients

AbstractBy the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.

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.

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