scholarly journals A Sharp Rellich Inequality on the Sphere

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 288
Author(s):  
Songting Yin
Keyword(s):  
Type Inequality ◽  
Best Constant ◽  
Rellich Inequality

We obtain a Rellich type inequality on the sphere and give the corresponding best constant. The result complements some related inequalities in recent literatures.

A note on the one-dimensional maximal function

1989 ◽  
Vol 111 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
Antonio Bernal
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Type Inequality ◽  
Best Constant ◽  
Weak Type Inequality ◽  
One Dimensional ◽  
Littlewood Maximal Function ◽  
The One

SynopsisIn this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

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 An estimate for the best constant in theLp-Wirtinger inequality with weights

2008 ◽  
Vol 6 (1) ◽  
pp. 1-16
Author(s):  
Raffaella Giova
Keyword(s):  
Type Inequality ◽  
Best Constant ◽  
Wirtinger Inequality

We prove an estimate for the best constantCin the following Wirtinger type inequality∫02πa|w|p≤C∫02πb|w′|p.

An Opial-type inequality with an integral boundary condition

2005 ◽  
Vol 461 (2060) ◽  
pp. 2635-2651 ◽  
Author(s):  
Richard C Brown ◽  
Michael Plum
Keyword(s):  
Boundary Condition ◽  
Type Inequality ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Best Constant ◽  
Classical Inequality

We determine the best constant K and extremals of the Opial-type inequality , where y is required to satisfy the boundary condition . The techniques employed differ from those utilized recently by Denzler to solve this problem, and also from those used originally to prove the classical inequality; but they also yield a new proof of that inequality.

On a Reverse of a Hardy-Hilbert Type Inequality

2012 ◽  
Vol 170-173 ◽  
pp. 2966-2969
Author(s):  
Bao Ju Sun
Keyword(s):  
Type Inequality ◽  
Constant Factor ◽  
Best Constant ◽  
Two Parameters ◽  
Best Constant Factor ◽  
Hilbert Type

In this paper, a reverse of Hardy-Hilbert type inequalities with a best constant factor is given by introducing two parameters.

scholarly journals On Spectrum of the Laplacian in a Circle Perforated along the Boundary: Application to a Friedrichs-Type Inequality

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
G. A. Chechkin ◽  
Yu. O. Koroleva ◽  
L.-E. Persson ◽  
P. Wall
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Boundary Value Problem ◽  
Unit Circle ◽  
Boundary Value ◽  
Type Inequality ◽  
Best Constant ◽  
Spectrum Of The Laplacian ◽  
Distance To The Boundary

In this paper, we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results, the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.

scholarly journals Improved Hardy-Rellich inequalities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Biagio Cassano ◽  
Lucrezia Cossetti ◽  
Luca Fanelli
Keyword(s):  
Magnetic Field ◽  
Hardy Inequality ◽  
Best Constant ◽  
Rellich Inequality ◽  
Free Case ◽  
Optimal Inequality ◽  
Free Operator ◽  
The Hardy Inequality

<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [<xref ref-type="bibr" rid="b21">21</xref>] for the Hardy inequality, later by Evans and Lewis in [<xref ref-type="bibr" rid="b9">9</xref>] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.</p>

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