Generalizations of Reid Inequality

A generalization of a half-discrete Hilbert's inequality

Considering some parameters and by means of an inequality of Hadamard, we derive general half-discrete Hilbert-type inequalities. Then we highlight some special cases.

Hardy-Littlewood-Pólya重排不等式的矩阵推广及其应用

A Matrix Generalization of the Hardy-Littlewood-Pólya Rearrangement Inequality and Its Applications

本文通过分析正交矩阵上的优化问题，将Hardy-Littlewood-Pólya重排不等式推广到正定矩阵与矩形矩阵，并据此推导了信号处理、机器学习领域类距离函数的新不等式。

Mercer不等式及其算子推广的若干形式

Some Generalizations of Mercer inequality and its operator extensions

本文研究了超二次函数的Mercer不等式及其算子推广，通过用正算子替换常数得到更一般的形式，并导出了Jensen算子不等式及Hermite-Hadamard型不等式。

A remark on a generalization of a logarithmic Sobolev inequality to the Holder class

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brezis-Gallouet-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but the wider class of Holder continuous functions.

Sharp Lyapunov's inequality for the measurable sets with infinite measure, with generalization to the Grand Lebesgue spaces

We extend the classical Lyapunov inequality on the measurable space with infinite measure and on the so-called Grand Lebesgue spaces (GLS). We find also the exact value for correspondent constant. Possible applications: Functional Analysis (for instance, interpolation of operators), Integral Equations, Probability Theory and Statistics (tail estimations for random variables) etc.

Sharp generalizations of the multilinear Bohnenblust--Hille inequality

We prove that the multilinear Bohnenblust--Hille is a particular case of a quite general family of optimal inequalities.

Generalization of two-dimensional Hardy type inequality for fuzzy integrals

In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis dealing with set-valued functions. The first one is given for a pseudo-integral of set-valued function where pseudo-addition and pseudo-multiplication are constructed by a monotone continuous function $g:[0, \infty ]\to[0, \infty]$. Another is given by the semiring $([0, 1], \max, \odot)$ where pseudo-multiplication is generated by an increasing continuous function.

A new generalization of Ostrowski type inequality on time scales

In this paper we first extend a generalization of Ostrowski type inequality on time scales for functions whose derivatives are bounded and then unify corresponding continuous and discrete versions. We also point out some particular integral type inequalities on time scales as special cases.

A generalization of Ostrowski inequality on time scales for k points

In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases.

A dual form of the sharp Nash inequality and its weighted generalization

The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the $L^2$ norm to the infimal convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also y…

Improvements and generalizations of two Hardy type inequalities and their applications to the Rellich type inequalities

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 best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.

Generalization of the Ehrling inequality and universal characterization of completely continuous operators

The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operators. Firstly, a modified notion of continuity for linear operators, named \emph{Ehrling continuity} and inspired by the classical Ehrling inequality, is introduced, and then, a necessary and sufficient condition for Ehrling continuity is provided via arguments based on general topology. Secondly, general completely continuous operators between normed spaces are characterized in terms of (generalized) Ehrling type ineq…

关于Cauchy-Schwarz-Bunyakovsky不等式的一般化及其在弹性力学中的应用

On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticity

本文系统阐述了CBS不等式的离散与积分形式、在n维欧几里得空间及其线性子空间中的推广及加强形式，并给出几何解释和关键证明，强调了其在弹性问题中的重要作用。

A generalization of the concept of distance based on the simplex inequality

We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, $$ where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ f…

θ-metric spaces: A generalization

In this paper, using a more generalized inequality instead of triangle inequality, the notion of θ-metric space is introduced. Some important properties of induced topology by such spaces are presented. Also, Banach and Caristi type fixed point in such spaces are proved.

On a generalization of Busemann's intersection inequality

Busemann's intersection inequality gives an upper bound for the volume of the intersection body of a star body in terms of the volume of the body itself. Koldobsky, Paouris, and Zymonopoulou asked if there is a similar result for $k$-intersection bodies. We solve this problem for star bodies that are close to the Euclidean ball in the Banach-Mazur distance. We also improve a bound obtained by Koldobsky, Paouris, and Zymonopoulou for general star bodies in the case when $k$ is proportional to the dimension.

Further generalizations, refinements, and reverses of the Young and Heinz inequalities

In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars. Applications to unitarily invariant norm inequalities involving positive semidefinite matrices are also given.

带局部权重的庞加莱-索伯列夫不等式的推广

Generalization for Poincaré--Sobolev inequalities with local weights

本文研究了不假设Muckenhoupt条件时，依赖区域立方体的局部权重下庞加莱-索伯列夫不等式的加权推广，并将其应用于齐次与非齐次型索伯列夫嵌入定理的加权推广。

A generalization of Costa's Entropy Power Inequality

Aim of this short note is to study Shannon's entropy power along entropic interpolations, thus generalizing Costa's concavity theorem. We shall provide two proofs of independent interest: the former by $Γ$-calculus, hence applicable to more abstract frameworks; the latter with an explicit remainder term, reminiscent of [20], allowing us to characterize the case of equality.

An Extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux's Generalization of the Sobolev Inequality to Continuous Dimensions

This paper extends a stability estimate of the Sobolev Inequality established by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality." Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb: "Is there a natural way to bound $\| \nabla φ\|_2^2 - C_N^2 \| φ\|_\frac{2N}{N-2}^2$ from below in terms of the 'distance' of $φ$ from the manifold of optimizers in the Sobolev Inequality?" Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other sh…