New integral inequalities via P-convexity
Wenjun Liu
本文推广了当绝对值的幂是$P-$凸函数时积分$\int_a^b (x-a)^p(b-x)^qf(x)dx$的一些新的估计。
In this note we extend some new estimates of the integral $\int_a^b (x-a)^p(b-x)^qf(x)dx$ for functions when a power of the absolute value is $P-$convex.
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New integral inequalities via $(α,m)$-convexity and quasi-convexity
本文分别建立了$(α, m)-$凸函数和拟凸函数的一些新的积分不等式。我们的结果在特殊情况下重新获得了已知的结果。
Inequalities via φ_{h,m}-convexity
本文定义了φ_{h，m}-凸函数，并证明了这类函数的几个不等式。
In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.
On some inequalities for different kinds of convexity
本文得到了关于n_{s}-凸函数、n_{s}-Godunova-Levin函数、n_{s}-P-函数和log-n_{s}-凸函数的几个不等式。最后，我们定义了凸-拟凸函数类，并研究了这类函数的一些性质。
Improvement of Jensen, Jensen-Steffensen's, and Jensen's functionals related inequalities for various types of convexity
本文针对一致凸、phi-凸和超二次函数，改进Jensen、Jensen-Steffensen及Jensen泛函相关不等式。
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
Generalisations of Integral Inequalities of Hermite-Hadamard type through Convexity
本文建立了与著名的Hermite-Hadamard积分不等式有关的几个可微映射的不等式，这些不等式是关于其导数为$s$ - $(α,m)$ -凸的映射的积分不等式的推广，所得的积分不等式给出了比已有的一些积分不等式更好的估计.然后将这些不等式应用于数值积分和一些特殊的方法。
Inequalities and higher order convexity
我们研究以下问题：给定n个实参数a1，...，an和n个实权重w1，...，wn，在什么条件下不等式w1 f（a1）+ w2 f（a2）+. + wn f（an）>= 0对于某些给定整数k具有非负k阶导数的所有函数f成立？利用简单的组合技巧，我们可以证明从Fuchs不等式到Schur凸性判据的许多定理的推广。
A fundamental paper of Elliott Lieb from 1973 has been the basis for much beautiful work on matrix inequalities by many people over the following years. We review a well-connected set of these developments. Some new proofs are provided.
On Iyengar-Type Inequalities via Quasi-Convexity and Quasi-Concavity
本文给出了涉及拟凸（拟凹）函数的Iyengar型不等式的一些新的估计。这些估计是最近获得的一些估计的改进。给出了梯形公式的一些误差估计。此外，还提供特殊手段的申请。
Interpolations, convexity and geometric inequalities
我们调查的一些相互作用的谱估计的Hörmander型，退化的蒙日-安培方程和几何不等式的对数相关的Brunn-Minkowski，Santaló或Busemann不等式。
We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results, we also give new inequalities improving previous known results. In particular, we show some inequalities for relative operator entropies and quantum mechanical entropies.
Symmetric norms and reverse inequalities to Davis and Hansen-Pedersen characterizations of operator convexity
给出了矩阵上对称范数的重排不等式以及算子凸函数的相关结果。
Convexity or concavity inequalities for Hermitian operators
给出了一些基本凸性不等式的矩阵形式。关于同一主题的进一步结果在最近关于arxiv：1的论文中得到了证明。埃尔米特算子和凸函数，2。对称范数的一个不等式，3.凸函数和对数凸函数的特征值不等式（与Jaspal Singh AUJLA合著）。
Employing the notion of operator log-convexity, we study joint concavity$/$ convexity of multivariable operator functions: $(A,B)\mapsto F(A,B)=h\left[ Φ(f(A))\ σ Ψ(g(B))\right]$, where $Φ$ and $Ψ$ are positive linear maps and $σ$ is an operator mean. As applications, we prove jointly concavity$/$convexity of matrix trace functions $\Tr\left\{ F(A,B)\right\}$. Moreover, considering positive multi-linear mappings in $F(A,B)$, our study of the joint concavity$/$ convexity of $(A_1,\cdots,A_k)\mapsto h\left[ Φ(f(A_1),\cdots,f(A_k))\right]$ provides some generalizations and complement to results o…
Simpson Type Inequalities via $φ$-Convexity
本文对导数绝对值为$φ$ -凸的函数得到了几个Simpson型不等式。
Means refinements via convexity
这篇文章的主要目标是找到一个凸函数和它的割线之间的精确差异，作为一个限制的积极的数量。这个想法将被表达为一个凸不等式，导致完善和逆转的既定的不等式处理不同的手段。这些不等式的意义是写一个不等式，汇集和完善几乎所有已知的不等式治疗算术，几何，调和和海因茨手段，为数字和运营商。
In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illustrate our work by providing a positive answer for the question addressed in [12] for the convexity of a certain matrix operator function. Moreover, and motivated by the results of A. Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt num…
Counterexamples for the convexity of certain matricial inequalities
这篇论文被撤回。我们在引理4.1中发现了一个错误
On Hadamard Type Inequalities Involving Several Kind of Convexity
本文不仅推广了Gill等人在[7]中关于对数凸函数的结果，而且得到了对数凸、m凸和（alpha，m）-凸函数的Hadamard型不等式.
Some convexity properties in direct integrals and Köthe-Bochner spaces
由Haydon、Levy和Raynaud在1991年引入的直接积分概念是著名的向量值函数的Köthe-Bochner空间概念的推广（使用一族目标空间而不是一个空间）。本文将讨论直接积分中的严格凸性、局部一致凸性和一致凸性等经典几何性质。我们还将考虑强凸和非常凸的Köthe-Bochner空间。
Displacement convexity of entropy and related inequalities on graphs
我们引入了有限图上概率测度集上的插值路的概念。利用这一概念，我们首先证明了熵沿这条路径的位移凸性，并在离散情形下得到了Prekopa-Leindler型不等式、Talagrand输运熵不等式、某些HWI型不等式以及log-Sobolev型不等式.为了说明通过例子，我们将我们的结果应用到完全图和超立方体，我们的结果是最佳的-通过传递到极限，我们恢复经典的log-Sobolev不等式的标准高斯测量与最佳常数。
Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of…
Some singular value inequalities via convexity
如果$c_1(Z) \geq ... \geq c_n(Z)$表示任何$n \times n$矩阵$Z,$的列向量的欧几里德长度，则与Hadamard乘积相关的基本不等式表明$$ \sum_{i=1}^k σ_i(X^*Y \circ B) \leq \sum_{i=1}^k c_i(X) c_i(Y) σ_i(B) \qquad 1 \leq k \leq n,$$，其中$σ_i(\cdot)$是$i$第奇异值。在本文中，我们将提供一个简单的证明，这一结果通过凸性的论点。此外，还利用这种技巧得到了一些更进一步的奇异值不等式。
On the role of Convexity in Functional and Isoperimetric Inequalities
这是我们以前工作0712.4092的延续。众所周知，各种等周不等式都蕴含着它们的函数"对应物“，但一般来说这不是等价的。我们表明，在一定的凸性假设下（例如，在欧几里得空间的对数凹概率测度），后者的含义实际上可以逆转非常一般的不等式，推广了反向形式的Cheeger不等式由于Buser和Ledoux。我们开发了一个连贯的单一框架之间的传递等周不等式，Orlicz-Sobolev功能不等式和容量不等式，后者是由Maz'ya和扩展的Barthe-Cattiaux-Roberto的概念。作为一个应用程序，我们扩展了已知的结果，由于后者作者的稳定性下张量化的等周轮廓，当没有中心极限障碍。作为另一个应用，我们证明了在凸性假设下，$q$ -Sobolev不等式（$q \in [1,2]$）等价于一类适当的等周不等式，推广了Bakry-Ledoux和Bobkov-Zegarlinski的结果.我们的结果推广到了更一般的满足Bakry-Emery的$CD(0,\infty)$曲率维数条件的具有密度的黎曼流形的情形.
Entropies, convexity, and functional inequalities
我们的目的是提供一个简短的和自我包含的合成，概括和统一各种相关和不相关的作品，涉及我们所谓的Phi-Sobolev功能不等式。这种不平等有关的Phi熵可以被视为特别是作为一个包容性的内插庞加莱和毛对数Sobolev不等式。除了已知的材料之外，还提供了扩展，并且针对某些方面给出了改进。稳定的张量积，卷积和有界扰动处理。我们证明了在φ上简单凸性假设下，这类不等式在许多情形下成立，包括超压缩扩散、一致严格对数凹测度、Wiener测度（黎曼流形上布朗运动的路径空间）和一般Poisson空间（包括某些纯跳Levy过程和相关无穷可分律的路径空间）.证明很简单，基本上依赖于凸性。我们最后以一个简短的类比结束，这个类比是受《动力学气体与信息论》中玻尔兹曼-香农熵的启发。
On the convexity of Berezin range and Berezin radius inequalities via a class of semi-norms
该研究引入新的$\u03c3_\u03bc$-Berezin半范数并推导相关不等式，以此改进有界线性算子的Berezin半径估计，同时研究两类空间上特定算子Berezin值域的凸性。
A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten $p$-classes for p>2 is proved
Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm
本文通过一类半范数研究了Berezin范围的凸性及Berezin半径不等式，改进了现有结果。
The quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. We prove Toeplitz-Hausdorff like theorem, that is, for any given quaternionic matrix every section of its quaternionic numerical range is convex. We provide some additional equivalent conditions for the quaternionic numerical range of matrices over quaternions to be convex and prove some numerical radius inequalities.
In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal sets $\mathbb{R}^α$ $(0<α\leq 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite-Hadamard (H-H) and Fejér-Hermite-Hadamard (Fejér-H-H) types inequalities. We also obtained a new result of the Fejér-H-H type for the function whose derivative in absolute value is the generalized $(h-m)$-convexity on fractal sets. Some applications to random variables and …
A simple proof for the convexity of the Choquet integral
本文给出了当相应的集函数是次模时Choquet积分凸性的一个初等而直接的证明。
Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space
本文利用EVI梯度流保持子流形不变的度量空间一般原理，研究Wasserstein空间子流形上能量与熵泛函的测地凸性，得到若干新凸性结果并改进Talagrand和HWI不等式。
Some new minimax theorems for generalized convexity
本文在非紧致设定下建立了扩展 Simons 极小极大定理等经典结果的双函数极小极大不等式，并证明单函数极小极大等式与 Simons 不等式等价，同时给出了相关应用。
In this paper, we introduce the notion of conditional $h$-convex functions and we prove an operator version of the Jensen inequality for conditional $h$-convex functions. Using this type of functions, we give some refinements for Ky-Fan's inequality, arithmetic-geometric mean inequality, Chrystal inequality, and H$\ddot{o}$lder-McCarthy inequality. Many of the other inequalities can be refined by applying this new notion.
Bellman VS Beurling: sharp estimates of uniform convexity for $L^p$ spaces
利用Bellman函数方法得到了经典的Hanner不等式.这些不等式给出了Lebesgue空间凸性模的精确估计。从微分几何的简单想法帮助我们找到贝尔曼函数既不使用“魔术猜测”也不计算。
Some Convexity Criteria for Differentiable Functions on the 2-Wasserstein Space
本文提出2-Wasserstein空间上可微函数测地凸性的新准则，通过无加速度曲线及Wasserstein梯度与Hessian不等式进行刻画。
On fractional inequalities via Montgomery identities integrals
利用Montgomery恒等式积分给出了几个新的Riemann-Liouville型分数次积分不等式.
Motivated by some recently established operator Jensen-type inequalities related to a usual convexity, in the present paper we derive several more accurate operator Jensen-type inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities for log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.
It is known that the volume functional $\,ϕ\mapsto\int e^{-ϕ}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a facto…
Unified treatment of fractional integral inequalities via linear functionals
本文证明了涉及两个保序线性泛函的几个不等式。本文研究了变界函数、Lipschitz和H\”older型函数等的不等式。这些结果为我们得到各种分数次积分算子如Riemann-Liouville分数次积分算子、Hadamard分数次积分算子、分数次超度量积分和相应的q-积分的不等式提供了一个简洁的方法。
Some integral inequalities via Caputo and Liouville fractional integral operators for m-convex functions
本文利用基本不等式和m-凸函数性质，获得了包含Caputo分数阶导数的新积分不等式，并给出了文献引理左边的上界。
Some Generalizations of Hermite-Hadamard type Integral Inequalities and their Applications
本文建立了与著名的Hermite-Hadamard积分不等式相联系的几类可微映射的不等式，其中Hermite-Hadamard积分不等式的导数是（h -（$α$; m））-凸的广义积分不等式给出了一些比已有的估计更好的估计。然后将这些不等式应用于数值积分和一些特殊的方法。
Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities
本文研究Fisher-Rao度量下的梯度流动力学，探讨测地凸性与函数不等式，提出不依赖目标分布log-concavity或log-Sobolev常数的分析方法，确保动力学收敛速率在一般分布下保持一致。
The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalities
本文研究Wasserstein Sobolev空间，证明柱函数在能量意义下稠密，并据此得出空间的自反性与一致凸性条件，以及Clarkson型不等式的充分条件。
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this p…
Hermitian operators and convex functions
给出了几个涉及凸组合和压缩的特征值不等式。这些不等式是基本凸性不等式f（（a+b）/2）<（f（a）+f（b））/2的矩阵形式.
Some integral inequalities for α-, m-, (α,m)-logarithmically convex functions
本文对不等式绝对值为α-，m-，（α，m）-代数凸的函数，利用初等不等式建立了几个新的Hadamard型不等式.
Weighted variation inequalities for differential operators and singular integrals in higher dimensions
利用$2<q<\infty$证明了高维微分算子和奇异积分算子的加权$q$ -变分不等式.并给出了这些不等式的向量值推广。
Hermite-Hadamard type inequalities for harmonically $(α,m)$-convex functions via fractional integrals
利用分数次积分建立了调和$(α,m)$ -凸函数的Hermite-Hadamard型不等式，并得到了这类函数的Hermite-Hadamard型不等式.
From Majorization to Horn and Unitary-Orbit Inequalities
本文提出一个统一框架，将优超不等式提升为完整的Horn不等式（等价于酉轨道不等式），并应用于三类算子不等式，得到推广的酉轨道不等式。
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