scholarly journals On Further Analogs of Hilbert's Inequality

2007 ◽  
Vol 2007 ◽  
pp. 1-6 ◽  
Author(s):  
Yongjin Li ◽  
You Qian ◽  
Bing He
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Hilbert’S Inequality ◽  
New Inequalities

By introducing the function|lnx−lny|/(x+y+|x−y|), we establish new inequalities similar to Hilbert's type inequality for integrals. As applications, we give its equivalent form as well.

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 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.

Improved Young and Heinz operator inequalities for unitarily invariant norms

2020 ◽  
Vol 70 (2) ◽  
pp. 453-466
Author(s):  
A. Beiranvand ◽  
Amir Ghasem Ghazanfari
Keyword(s):  
Hilbert Space ◽  
Type Inequality ◽  
Young Inequality ◽  
Unitarily Invariant Norm ◽  
Operator Inequalities ◽  
Schmidt Norm ◽  
Unitarily Invariant Norms ◽  
Invariant Norm ◽  
Kantorovich Constant ◽  
New Inequalities

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn(ℂ): $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r0 = min{ν, 1 – ν}.

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.

Some refined Young type inequalities using different weights

2021 ◽  
Author(s):  
Leila Nasiri ◽  
Bahman Askari
Keyword(s):  
Type Inequality ◽  
Matrix Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Unitarily Invariant Norms ◽  
New Inequalities

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

scholarly journals On a Generalization of Hilbert-Type Integral Inequality

2008 ◽  
Vol 2008 ◽  
pp. 1-8
Author(s):  
Sun Baoju
Keyword(s):  
Integral Inequality ◽  
Equivalent Form ◽  
Type Inequality ◽  
Type Integral ◽  
Hilbert Type

By introducing some parameters, we establish generalizations of the Hilbert-type inequality. As applications, the reverse and its equivalent form are considered.

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