Related papers: The optimal constant in Hardy-type inequalities
In this article we establish new improvements of the optimal Hardy inequality in the half space. We first add all possible linear combinations of Hardy type terms thus revealing the structure of this type of inequalities and obtaining best…
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued…
A refinement of the Hardy inequality has been presented by use of superquadratic function.
The best constant in the usual Lp norm inequality for the centered Hardy-Littlewood maximal function on R1 is obtained for the class of all ``peak-shaped'' functions. A positive function on the line is called ``peak-shaped'' if it is…
We study some Hardy-type inequalities involving a general norm in $R^n$ and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.
The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n)…
This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric…
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to…
We present a review of results that have been obtained in the past twenty-five years concerning the $L^p$-Hardy inequality with distance to the boundary. We concentrate on results where the best Hardy constant is either computed exactly or…
Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the…
The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n…
For absolutely convergent series we state explicitly a one-sided summation estimate that can be viewed as the discrete analogue of the change of variable formula on the half line. This estimate is implicit in Pascal Lef\`evre's recent…
We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a…
We present a unified approach to improved $L^p$ Hardy inequalities in $\R^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where distance is…
We compute the best constant in functional integral inequality called the Hardy-Leray inequalities for solenoidal vector fields on $\mathbb{R}^N$. This gives a solenoidal improvement of the inequalities whose best constants are known for…
We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the…
We show that the sharp constant in the classical $n$-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for $n=2$ without the axisymmetry…
We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with…
The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy-Rellich…
We study Hardy inequalities for antisymmetric functions in three different settings: euclidean space, torus and the integer lattice. In particular, we show that under the antisymmetric condition the sharp constant in Hardy inequality…