Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

Some quantum estimates of Hermite-Hadamard inequalities for convex functions

In this study, based on a new quantum integral identity, we establish some quantum estimates of Hermite-Hadamard type inequalities for convex functions. These results generalize and improve some known results given in literatures.     

Generalization of Popoviciu-Type Inequalities Via Fink’s Identity

We obtained useful identities via Fink’s identity, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the ?eby?ev functional. Some results relating to the Grüss- and Ostrowski-type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functional associated with the obtained inequalities.     

Integral inequalities for coordinated Harmonically convex functions

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.     

Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

Asymptotical good behavior on inequalities with completely approximate K-T concept

We characterize a wide class of regular convex functionals that are asymptotically well behaved on a convex set given by (infinite) inequalities, namely, those restricted functions whose stationary sequences (bounded or not) are minimizing ones. After showing the equivalence with the Kuhn-Tucker type stationarity, we prove that the class of such functions remains unchanged when the Kuhn-Tucker system is completely relaxed. This allows us to proceed for enlarging the scope of convergence of certain penalty (exterior as well as interior) methods including a new exterior penalization for infinite inequalities.     

另一个新的与Hadamard不等式相关的映射

对于凸函数的Hadamard不等式的推广形式的不等式,本文引进了另一个与这个不等式相关的映射,从而给出了该不等式连续的加细.同时也提及了它的某些应用.     

Weighted Hermite–Hadamard–Mercer type inequalities for convex functions

In this paper, first, we prove the weighted Hermite–Hadamard–Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right‐sides of weighted Hermite–Hadamard–Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works.     

Valid inequalities for separable concave constraints with indicator variables

We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs.     

Sharp discrete inequalities and applications to discrete variational problems

Some new discrete inequalities involving monotonic or convex functions are obtained. While these are interesting inequalities in their own right, they can be applied to solving certain types of discrete variational problems effectively.     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

Inequalities for convex functions via Stieltjes integral

We present a method of proving inequalities for convex functions with use of Stieltjes integral. First, we show how some well-known inequalities can be obtained, and then we show how new inequalities and stronger versions of some existing results can be obtained.     

Multidimensional Hermite-Hadamard inequalities and the convex order

The problem of establishing inequalities of the Hermite-Hadamard type for convex functions on n-dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order. We present two results of partial generality which unify and extend the most part of the multidimensional Hermite-Hadamard inequalities existing in the literature, at the same time that lead to new specific results. The first one fairly applies to the most familiar kinds of polytopes. The second one applies to symmetric random vectors taking values in a closed ball for a given (but arbitrary) norm on Rn. Related questions, such as estimates of approximation and extensions to signed measures, also are briefly discussed.     

Divergence for s-concave and log concave functions

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré inequalities for such functions. This leads naturally to the concept of f-divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valuation property. Applications are given in the theory of convex bodies.     

New information-entropic relations for Clebsch–Gordan coefficients

Using properties of the Shannon and Tsallis entropies, we obtain new inequalities for the Clebsch–Gordan coefficients of the group SU(2). For this, we use squares of the Clebsch–Gordan coefficients as probability distributions. The obtained relations are new characteristics of correlations in a quantum system of two spins. We also find new inequalities for Hahn polynomials and the hypergeometric functions ₃F₂.     

On a class of discontinuous vector-valued functions and the associated quasi-variational inequalities

We study in detail a class of discontinuous vector-valued functions defined on a closed convex subset of Rn, which was introduced by B. Ricceri [7] and which is very useful in the theory of variational inequalities. The results are used to give a new proof for the existence theorem due to P. Cubiotti [3]. The proof allows us to have a better understanding of quasi-variational inequalities associated with the abovementioned class of functions.     

On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems,including inequalities for log concave functions,and with an application to the diffusion equation

We extend the Prékopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prékopa-Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prékopa-Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.     

Fejer and Hermite-Hadamard type inequalities for superquadratic functions

In this paper we use basic properties of superquadratic functions to obtain new inequalities including Fejer's type and Hermite-Hadamard type inequalities. For superquadratic functions which are also convex, we get refinements of known results.     

一个新的与Hadamard不等式相关的映射

对于一个最近发表的凸函数的Hadamard不等式的推广形式的不等式,本文引进了—个与这个不等式相关的的映射,从而给出了该不等式连续的加细.同时提及了它的某些应用.