Gang Lu, Wenlong Sun, Hanyue Qiao, Yuanfeng Jin, Choonkil Park
In this paper, we introduce and investigate general 3-variable Jensen $ρ$-functional equation, and prove the Hyers-Ulam stability of the Jensen functional equations associated with the general 3-variable Jensen $ρ$-functional inequalities in complex Banach spaces.
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Using fixed point methods, we prove the generalized Hyers--Ulam--Rassias stability of ternary homomorphisms, and ternary multipliers in ternary Banach algebras for the Jensen--type functional equation $$f(\frac{x+y+z}{3})+f(\frac{x-2y+z}{3})+f(\frac{x+y-2z}{3})= f(x).$$
We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra. Reverses and refinements of Jensen's and Slater's type inequalities are provided. Some examples for particular convex functions of interest are given as well.
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。
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.
Jensen-type inequalities on pairs of measure spaces and their applications
本文提出正测度空间对上的Jensen型不等式框架，导出积分算子、Fejér均值的范数与熵估计及概率矩界，并给出擦除鲁棒性结果，统一了多个数学领域。
Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti…
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.
In this paper, we state some characterizations of $h$-convex function is defined on a convex set in a linear space. By doing so, we extend the Jensen-Mercer inequality for $h$-convex function. We will also define $h$-convex function for operators on a Hilbert space and present the operator version of the Jensen-Mercer inequality. Lastly, we propound the complementary inequality of Jensen's inequality for $h$-convex functions.
In this paper, we provide some inequalities for $P$-class functions and self-adjoint operators on a Hilbert space including an operator version of the Jensen's inequality and the Hermite-Hadamard's type inequality. We improve the Hölder-MacCarthy inequality by providing an upper bound. Some refinements of the Jensen type inequality for $P$-class functions will be of interest.
We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in an earlier paper on non-regularity for Banach function algebras. We show that, even for the natural uniform algebras $R(X)$ (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular, but such that $R(X)$ has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Ou…
In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability of a Cauchy-Jensen additive functional equation in various normed spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
Stability of involutions on Banach Algebra by fixed point method
该研究采用Jensen函数方程稳定性的不动点方法，证明巴拿赫代数上对合的Hyers-Ulam-Rassias稳定性与超稳定性，还得到使带近似对合的巴拿赫代数成为$C^*$-代数的条件。
Let $f$ be an operator convex function on $(0,\infty)$, and $Φ$ be a unital positive linear maps on $B(H)$. we give a complementary inequality to Davis-Choi-Jensen's inequality as follows \begin{equation*} f(Φ(A))\geq \frac{4R(A,B)}{(1+R(A,B))^2}Φ(f(A)), \end{equation*} where $R(A, B)=\max\{r(A^{-1}B) ,r(B^{-1}A)\}$ and $r(A)$ is the spectral radius of $A$. We investigate the complementary inequalities related to the operator power means and the Karcher means via unital positive linear maps, and obtain the following result: If $A_{1}$, $A_{2}$,\dots, $A_{n}$, are positive definite operators in…
A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.
We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty) \to \mathbb{R}$ is a continuous convex function with $f(0)\leq 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-δ_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$, where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $δ_f = f(0)+f(M) - 2 f(\frac{M}{2})$.
In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.
We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence spaces. Several inequalities appear to be new even in the finite dimensional case.
We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.
The main aim of this work is to give a general approach to the celebrated Kahane-Salem-Zygmund inequalities. We prove estimates for exponential Orlicz norms of averages $\sup_{1\le j \leq N} \big |\sum_{1 \leq i \leq K}γ_i(\cdot) a_{i,j}\big|$ where $(a_{i,j})$ denotes a matrix of scalars and the $(γ_i)$ a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get novel Kahane-Salem-Zygmund type inequalities -- in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spac…
We prove an abstract interpolation theorem which interpolates the (r,2)-summing and (s,2)-mixing norm of a fixed operator in the image and the range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett-Carl inequalities for identities acting between Minkowski spaces l_u as well as their analogues for Schatten classes S_u. Furthermore, our techniques motivate a study of Bennett-Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals.
Some Properties of Mordukhovich Derivatives with Applications to Locally Variational Inequalities in Banach Spaces
本文研究Banach空间中Frechet导数与Mordukhovich导数的联系，发现集值映射导数性质，引入局部变分不等式概念，并揭示其解与Mordukhovich导数的关联。
Vector-Valued Hardy--Littlewood Maximal Inequalities on Ball Banach Function Spaces
本文通过向量值极大不等式刻画非中心哈代-利特尔伍德极大算子在球巴拿赫函数空间及其Köthe伴随空间上的有界性，所得结果补充了Nieraeth的近期研究成果。
Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra
本研究推广了Mond-Pecaric（MP）算子不等式方法，放宽凸函数与线性映射限制，推导超复函数新不等式，应用于函数逼近与界代数构建。
We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincaré-type polynomials. It turns out that the fulfillment of most of types of decoupling inequalities may depend on the geometry of Banach space.
Let $p$ be a positive number and $h$ a function on $\mathbb{R}^+$ satisfying $h(xy) \ge h(x) h(y)$ for any $x, y \in \mathbb{R}^+$. A non-negative continuous function $f$ on $K (\subset \mathbb{R}^+)$ is said to be {\it operator $(p,h)$-convex} if \begin{equation*}\label{def} f ([αA^p + (1-α)B^p]^{1/p}) \leq h(α)f(A) +h(1-α)f(B) \end{equation*} holds for all positive semidefinite matrices $A, B$ of order $n$ with spectra in $K$, and for any $α\in (0,1)$. In this paper, we study properties of operator $(p,h)$-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We …
We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving …
We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical Hölder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.
We establish variant Khintchine inequalities on normed spaces of Hanner type and cotype, in which the Rademacher distribution corresponding to classical Khintchine inequality is replaced by general symmetric distributions. The proof involves the $p$-barycenter and Birkhoff's ergodic theorem. More importantly, by employing these Khintchine inequalities, we get some lower bounds for Banach-Mazur distance between $l^p$-ball and a general centrally symmetric convex body.
In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double …
We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian …
Graph Convex Hull Bounds as generalized Jensen Inequalities
本文通过研究函数图像的凸包，为任意域上的任意函数给出期望的精确上下界，将 Jensen 不等式推广到有限/无限维及条件期望、Markov 算子情形。
Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$ is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if $E$ is a set of interpolation ($I_0$-set). However for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if $X$ is restricted to be ``natural'' then the result holds for all Sidon sets. We also cons…
Functional inequalities in the framework of Banach spaces
本研究将C. Schötz在Hilbert空间下的四边形不等式推广到Banach空间框架，基于优超理论和平行四边形法则替代形式推导，还得到若干拓展经典结论的泛函不等式。
Inequalities for Pairs of Measure Spaces and Applications
本文在一般乘积测度框架下建立了一种Jensen型不等式，扩展了Hölder和Minkowski等经典不等式，并应用于幂均、熵型和擦除鲁棒不等式以及卷积型算子和加权离散模型。
In this paper, combined with the P-angle function of Banach spaces and the geometric constants that can characterize Hilbert spaces, the new angular geometric constant is defined. Firstly, this paper explores the basic properties of the new constant and obtains some inequalities with significant geometric constants. Then according to the derived inequalities, this paper studies the relationship between the new constant and the geometric properties of Banach spaces. Furthermore, the necessary and sufficient condition for uniform non-squareness, and the sufficient conditions for uniform convexit…
We study a class of quasi-variational inequality problems defined over infinite dimensional Banach space and deduce sufficient conditions for ensuring solutions to such problems under the upper semi-continuity and pseudomonotonicity assumptions on the map defining the inequalities. The special structure of the quasi-variational inequality enables us to show the occurrence of solutions for such inequalities based on the classical existence theorem for variational inequalities. This special type of quasi-variational inequalities is motivated by the pure exchange economic problems and Radner equi…
Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u, v)-cocoercive mappings and v-strongly monotone mappings from Hilbert spaces to Banach spaces. Then we prove the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and improves [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (201…
Banach bimodule-valued positive maps: Inequalities and induced representations
本文研究了Banach双模值正映射的诱导表示，并推导了相关不等式。
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_μ^\infty$-spaces instead.
In this paper we introduce the quadratic weighted geometric mean for invertible elements x, y in a Hermitian unital Banach star-algebra and provide some inequalities for this mean under various assumptions for the elements involved.
We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of nonnegative matrices that define operators on Banach sequences we obtain additional results. Our results extend results of several authors that appeared relatively recently.
The existence of a solution, convergence and stability of the penalty method for variational inequalities with nonsmooth unbounded uniformly and properly monotone operators in Banach spase $B$ are investigated. All the objects of the inequality - the operator A, "the right-hand part" $f$ and the set of constrains $Ω$ - are to be perturbed. The stability theorems are formulated in terms of geometric characteristics of the spaces $B$ and $B^*$. The results of this paper are continuity and generalization of the Lions' ones, published earlier in \cite{l}. They are new even in Hilbert spaces.
Some inequalities for log-convex functions of selfadjoint operators on quaternionic Hilbert spaces
本文针对四元数希尔伯特空间上的自伴算子，利用对数凸函数证明了若干Jensen型不等式，并通过特定对数凸函数得到算子不等式的特殊情况，丰富了该领域的不等式研究。
In this paper we show the weak Banach-Saks property of the Banach vector space $(L_μ^p)^m$ generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty,$ where $m$ is any given natural number. When $m=1,$ this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of the space $(L_μ^p)^m$ for $1\leq p<+\infty$ and inequalities when these vector-valued functions are in …
These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.
We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebra, and in particular, in the associated Wiener algebra.
Maximal Ergodic Inequalities for Banach Function Spaces
本文推广 Transfer Principle 到 σ-紧局部紧 Hausdorff 群的保测作用，刻画 Banach 函数空间性质对弱型极大不等式的影响，并推出广泛遍历平均的几乎必然逐点收敛。
In this paper we introduce the notion of weak 2-positivity and present some examples. We establish some operator Cauchy--Schwarz inequalities involving the geometric mean and give some applications. In particular, we present some operator versions of Hua's inequality by using the Choi--Davis--Jensen inequality.
We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.
We establish Hölder type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.
We show that for quasi-greedy bases in real Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a constant which depends on the democracy functions and the quasi-greedy constant of the basis.
Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the $K$-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of $H^\infty$-functional calculus to this general setting.
Inequalities on the generalized and the joint spectral radius of Hadamard products
该研究针对Banach函数与序列空间上正算子有界集，证明了其Hadamard乘积的广义及联合谱半径新不等式，还针对正核算子给出相关不等式，推广改进了近期已有结果。
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1…
Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring decoupling inequalities with good constants depending only exponentially on the degree of the polynomial. Assuming the Banach space has finite cotype we achieve this for classical decoupling inequalities that compare the polynomials with their associate…
Let $0<p<q\leq\infty$ and $α\in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,α),L_q(0,α))$ in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.
In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{$\star$} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E γ([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ γ([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general U…
This paper presents a survey of maximal inequalities for stochastic convolutions in $2$-smooth Banach spaces and their applications to stochastic evolution equations.
Discrete logarithmic Sobolev inequalities in Banach spaces
本文在有限余型Banach空间上建立了离散超立方体上与维数无关的向量值Lp对数Sobolev不等式，并利用该不等式给出Hamming立方体非线性商的双Lipschitz畸变新下界。
The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,π)=0$ for every isometric representation $π$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,π)$ vani…
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