Wenjun Liu, Quoc Anh Ngo
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.
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In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case.
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 generalisation of inner product spaces of an inequality due to Ostrowski and applications for sequences and integrals are given.
In this paper we establish a refinement of the companion of Ostrowski inequality for functions of bounded variation. Applications for the trapezoid inequality, the mid-point inequality, and to probability density functions are also given.
Mean value theorem for quantum integral operator with application to sharp Ostrowski inequality
本文为量子微积分推导了拉格朗日中值定理，否证了文献中的Ostrowski不等式，提出并证明了一个尖锐的新不等式，并给出了中点不等式。
In this paper, we obtain some companions of Ostrowski type inequality for absolutely continuous functions whose second derivatives absolute value are s-convex and s-concave.
A companion of Ostrowski like inequality for mappings whose second derivatives belong to $L^{\infty}$ spaces is established. Applications to composite quadrature rules, and to probability density functions are also given.
In this paper, we obtain some companions of Ostrowski type inequality for absolutely continuous functions whose second derivatives absolute value are convex and concave.Finally, we gave some applications for special means.
In this paper we establish some new bounds for the companion of Ostrowski's inequality for the case when $f'\in L^1[a,b]$, $f"\in L^2[a,b]$ and $f'\in L^2[a,b]$, respectively. We point out that the results in the first and third cases are sharp and that some of these new estimations can be better than the known results. Some applications to composite quadrature rules, and to probability density functions are also given.
New Generalization of Perturbed Ostrowski Type Inequalities and Applications
本文建立了Lipschitz型函数的Ostrowski型不等式的推广，并给出了在数值积分和累积分布函数中的应用。
In this paper we obtain some weighted generalizations of Ostrowski type inequalities on time scales involving combination of weighted Δ-integral means, i.e., a weighted Ostrowski type inequality on time scales involving combination of weighted Δ-integral means, two weighted Ostrowski type inequalities for two functions on time scales, and some weighted perturbed Ostrowski type inequalities on time scales. We also give some other interesting inequalities and recapture some known results as special cases.
We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted mean of these n points is always greater or equal to a convex combination of the values of the function at some other weighted means of these points. Here, the left hand side contains only one weighted mean, while the right hand side may contain as many as possible, as long as there are finitely many. The weighted mean on the left hand side must have positive weights, while those on the right hand sid…
We prove a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-Δ)^{α/2}$, where $α\in (0,2)$.
In this note we generalize the trace inequality derived by [1] to the case where the number of terms of the sum (denoted by K) is arbitrary.
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.
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。
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。
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.
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.
We prove that the multilinear Bohnenblust--Hille is a particular case of a quite general family of optimal inequalities.
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.
In this paper, we improve the famous Reid Inequality related to linear operators. Some monotony results for positive operators are also established with a different approach from what is known in the existing literature. Lastly, Reid and Halmos-Reid inequalities are extended to unbounded operators.
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…
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…
Let $A,\;B$ be the positive semidefinite matrices. A matrix version of the famous Powers-Størmer's inequality $$2Tr(A^αB^{1-α})\geq Tr(A+B-|A-B|),\;\;\;0\leqα\leq 1,$$ was proven by Audenaert et. al. We establish a comparison of eigenvalues for the matrices $A^αB^{1-α}$ and $A+B-|A-B|, \; 0 \leq α\leq 1,$ subsuming the Powers-Størmer's inequality. We also prove several related norm 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.
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.
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.
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。
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…
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.
We present a new operator equality in the framework of Hilbert $C^*$-modules. As a consequence, we get an extension of the Euler--Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung-J.E. Pecaric and F. Zhang.
A Generalization of the Böttcher-Wenzel inequality for three rectangular matrices
本文针对三个矩形矩阵定义广义换位子ABC-CBA，估计其Frobenius范数并得到推广Böttcher-Wenzel不等式的结果，且在一维情形给出更紧上界。
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。
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.
We will formulate and prove a generalization of the isoperimetric inequality in the plane. Using this inequality we will construct an unitary space - and in consequence - an isomorphic copy of a separable infinite dimensional Hilbert space, which will appear also an RHKS.
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…
In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C g(f(x))dx \leq \frac12 \int_{-1}^1g(f(0)(1+t))dt, \] where |C| denotes the volume of C. If g? is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the…
We establish an operator extension of the following generalization of Bohr's inequality, due to M.P. Vasić and D.J. Kečkić: $$|\sum_{i=1}^n z_i|^r \leq (\sum_{i=1}^n α_i^{1/(1-r)})^{r-1}\sum_{i=1}^n α_i|z_i|^r \quad (r>1, z_i \in{\mathbb C}, α_i>0, 1 \leq i \leq n) .$$ We also present some norm inequalities related to our noncommutative generalization of Bohr's inequality.
Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombiari and Boas--Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert $C^*$-modules.
An Operator-Valued Haagerup Inequality for Hyperbolic Groups
研究超几何群上的算子值Haagerup不等式，改进Buchholz的结果，用单词分解关联的矩阵范数替代原不等式中的ℓ²范数。
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