scholarly journals On Hilbert's Inequality for Double Series and Its Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
Zhou Yu ◽  
Gao Mingzhe
Keyword(s):  
Real Function ◽  
Hardy Inequality ◽  
Double Series ◽  
Cauchy Inequality ◽  
Hilbert Inequality ◽  
Hilbert’S Inequality ◽  
Riesz Inequality

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real functionu(x)and a parameterλ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHpfunction are given.

scholarly journals A sharper form of half-discrete Hilbert inequality related to Hardy inequality

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6733-6740
Author(s):  
Al-Oushoush Nizar ◽  
L.E. Azar ◽  
Ahmad Bataineh
Keyword(s):  
Hardy Inequality ◽  
Hilbert Inequality ◽  
The Hardy Inequality

We introduce some new sharper forms of the half-discrete Hilbert inequality which are connected to the Hardy inequality.

scholarly journals On new strengthened Hardy-Hilbert's inequality

1998 ◽  
Vol 21 (2) ◽  
pp. 403-408 ◽  
Author(s):  
Bicheng Yang ◽  
Lokenath Debnath
Keyword(s):  
Weight Coefficient ◽  
Hilbert Inequality ◽  
Hilbert’S Inequality ◽  
Strengthened Version

In this paper, a new inequality for the weight coefficientω(q,n)in the formω(q,n):=∑m=1∞1m+n(nm)1/q              <πsin(π/p)−12π1/p+n−1/q(q>1,1p+1q=1,n∈N)is proved. This is followed by a strengthened version ofthe Hardy-Hilbert inequality.

On an Extension of Hilbert Inequality

2012 ◽  
Vol 166-169 ◽  
pp. 3023-3026
Author(s):  
Bao Ju Sun
Keyword(s):  
Cauchy Inequality ◽  
Hilbert Inequality ◽  
Hadamard Inequality

In this paper, by using Hadamard inequality and Cauchy inequality, an extension of Hilbert inequality is established.

On an extension of the Hardy-Hilbert theorem

2005 ◽  
Vol 42 (1) ◽  
pp. 21-35 ◽  
Author(s):  
J. Weijian ◽  
G. Mingzhe ◽  
G. Xuemei
Keyword(s):  
Beta Function ◽  
Summation Formula ◽  
Hilbert’S Inequality ◽  
Two Parameters

A weighted Hardy-Hilbert’s inequality with the parameter λ of form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\frac{{a_m b_n }}{{(m + n)^\lambda }}} < B^* (\lambda )\left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } a_{a_n }^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } b_n^q } } \right)^q }$$ \end{document} is established by introducing two parameters s and λ, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{p} + \tfrac{1}{q} = 1,p \geqq q > 1,1 - \tfrac{q}{p} < \lambda \leqq 2,B^* (\lambda ) = B(\lambda - (1 - \tfrac{{2 - \lambda }}{p}),1 - \tfrac{{2 - \lambda }}{p})$$ \end{document} is the beta function. B *(λ) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.

scholarly journals Heat kernels of the discrete Laguerre operators

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Aleksey Kostenko
Keyword(s):  
Hardy Inequality ◽  
Jacobi Polynomials ◽  
Heat Kernels ◽  
The Other ◽  
Heat Semigroup ◽  
Stochastic Completeness ◽  
Markovian Semigroup ◽  
Other Hand ◽  
The One

AbstractFor the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

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.

On a Refinement of Hardy-Hilbert Inequality

2012 ◽  
Vol 166-169 ◽  
pp. 3027-3030
Author(s):  
Bao Ju Sun
Keyword(s):  
Weight Coefficient ◽  
Hilbert Inequality ◽  
Maximum Minimum

In this paper, by using Maximum-minimum monotonic theorem and estimating the weight coefficient, a refinement of Hardy- Hilbert inequality is established.

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