The Hadamard determinant inequality — Extensions to operators on a Hilbert space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some ‘Fischer-type’ inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on $[0,\infty )$. We place emphasis on documenting necessary and sufficient conditions for equality to hold.     

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ? ^q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ? ^q (𝒫) for some q ≥ 1.     

Type,Cotype Constants and Clarkson's Inequalities for Banach Spaces

It is shown that a Banach space satisfies Clarkson's inequalities if and only if its “type or cotype constant” is 1, which implies in particular that the notions of G_o - and G_n - Fourier type by Milman [16] are equivalent. A sequence of related results is also given.     

Fej\''{e}r type inequalities for Harmonically-convex functions with applications

In this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. By using the notion of harmonic symmetricity, harmonic convexity and some auxiliary results, some new Fej\''{e}r type integral inequalities are presented. Applications to special means of positive real numbers are given as well.     

A stochastic calculus for continuous N-parameter strong martingales

Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale. Burkholder's inequalities prove to be an adequate tool to control the quadratic oscillations of M and the integral processes associated with it (i.e. multiple 1-stochastic integrals with respect to M and its quadratic variation) such that a 1-stochastic calculus for M can be designed. As the main results of this calculus, several Ito-type formulas are established: one in terms of the integral processes associated with M, another one in terms of the so-called ‘variations’, i.e. stochastic measures which arise as the limits of straightforward and simple approximations by Taylor's formula; finally, a third one which is derived from the first by iterated application of a stochastic version of Green's formula and which may be the strong martingale form of a prototype for general martingales.     

Norm inequalities for the spectral spread of Hermitian operators

In this work, we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite-dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the arithmetic–geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).     

On an upper bound of Sanov's inequality for the probability of a large deviation

In this paper a sharp upper bound of Sanov's inequality for the large probability of large deviations under the multinomial distribution is obtained. Examples of computing the upper bounds for the probabilities of misclassification in discriminant analysis are also given.     

Wang's Harnack inequalities for space–time white noises driven SPDEs with two reflecting walls and their applications

In this paper, we establish Wang's Harnack inequalities for Gaussian space–time white noises driven the stochastic partial differential equation with double reflecting walls, which is of the infinite dimensional Skorokhod equation. We first establish both the Harnack inequality with power and the log-Harnack inequality for the special case of additive noises by the coupling approach. Then we investigate the log-Harnack inequality for the Markov semigroup associated with the reflected SPDE driven by multiplicative noises using the penalization method and the comparison principle for SPDEs. As their applications, we study the strong Feller property, uniqueness of invariant measures, the entropy-cost inequality, and some other important properties of the transition density.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

On the information about individual utilities used in social choice

This paper investigates the consequences of the requirement tha a social choice method should convey as much information as possible about the individuals' values. It is first shown that if interpersonal comparisons of utilities are excluded, then the only information about individual values that can be used in the context of social choice is ordinal information. It is then argued that Arrow's independence condition demands that only a part of the information about the individual preferences be used in the social choice. Finally, it is shown that the requirement of maximal information gives strong support to Smith's (1973) ‘separability’ condition. In combination with Smith's result this shows that the choice methods that convey the most information about individual preferences are the ‘generalized point systems’ (also called ‘scoring functions’ or ‘ranking systems’).     

Functional inequalities for the Wright functions

In this paper, our aim is to show some mean value inequalities for the Wright function, such as Turán-type inequalities, Lazarevi?-type inequalities, Wilker-type inequalities and Redheffer-type inequalities. Moreover, we prove monotonicity of ratios for sections of series of Wright functions, the result is also closely connected with Turán-type inequalities. In the end of the paper, we present some other inequalities for the Wright function.     

A simultaneous projections method for linear inequalities

An iterative method for solving general systems of linear inequalities is considered. The method, a relaxed generalization of Cimmino's scheme for solving linear systems, was first suggested by Censor and Elfving. Each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities. The algorithm is particularly suitable for implementation on computers with parallel processors. We prove convergence from any starting point for both consistent and nonconsistent systems (to a feasible point in the first case, and to a weighted least squares type solutions in the second).     

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACT

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.     

Generalizations of Bohr's inequality in Hilbert C*-modules

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. Pe?ari? and F. Zhang.     

Lyapunov type inequalities for a fractional two‐point boundary value problem

In this paper, new Lyapunov‐type inequalities are obtained for the case when one is dealing with a class of fractional two‐point boundary value problems. As an application of this result, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

Bounds for Csiszár divergence and hybrid Zipf‐Mandelbrot entropy

The purpose of this paper is to give a series of inequalities of the Jensen type and their applications for Csiszár divergence. By using these results, we give many estimations for hybrid Zipf‐Mandelbrot entropy.     

The Janson inequalities for general up‐sets

Janson and Janson, ?uczak and Ruciński proved several inequalities for the lower tail of the distribution of the number of events that hold, when all the events are up‐sets (increasing events) of a special form—each event is the intersection of some subset of a single set of independent events (i.e., a principal up‐set). We show that these inequalities in fact hold for arbitrary up‐sets, by modifying existing proofs to use only positive correlation, avoiding the need to assume positive correlation conditioned on one of the events. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 391–395, 2015     

一类新的Radon型不等式及其应用

给出一类新的R adon型不等式,它们在代数不等式研究中有着广泛的应用,利用它们可直接得到一大批新的分式型不等式,也可运用它们证明或推广许多不等式.     

关于Hardy-Hilbert型不等式的新改进

利用加强了的H lder′s不等式及Eu ler-M aclaurin求和公式对两个带参数的H ardy-H ilbert型不等式作了改进,建立了一些新的不等式.