On weighted Hermite-Hadamard inequalities

In this paper, we give a weighted form of the Hermite-Hadamard inequalities. Some applications of them are also derived. The results presented here would provide extensions of those given in earlier works. Finally we pose two interesting problems.     

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.     

Generalized convexity and inequalities

Let R₊=(0,∞) and let M be the family of all mean values of two numbers in R₊ (some examples are the arithmetic, geometric, and harmonic means). Given m₁,m₂∈M, we say that a function is (m₁,m₂)-convex if f(m₁(x,y))?m₂(f(x),f(y)) for all x,y∈R₊. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m₁,m₂)-convexity on m₁ and m₂ and give sufficient conditions for (m₁,m₂)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.     

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.     

Optimality via generalized approximate convexity and quasiefficiency

In this article, we look beyond convexity and introduce the four new classes of functions, namely, approximate pseudoconvex functions of type I and type II and approximate quasiconvex functions of type I and type II. Suitable examples illustrating the non emptiness of the newly defined classes and distinguishing them from the existing classical notions of pseudoconvexity and quasiconvexity are provided. These newly defined concepts are then employed to establish sufficient optimality conditions for the quasi efficient solutions of a vector optimization problem.     

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.     

Generalized convexity and inequalities involving special functions

In this paper, the authors show a relation between the generalized convexity and super- (sub-)multiplicative property, and discuss some generalized convexity and inequalities involving the Gaussian hypergeometric function, the generalized η-distortion function and the generalized Grötzsch function μa(r).     

Some inequalities of Hermite-Hadamard type for s-convex functions

In this paper several inequalities of the left-hand side of Hermite-Hadamard’s inequality are obtained for s-convex functions.     

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x₁,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ?>0 such that if X=(X₁,…,Xg) is a g-tuple of n×n symmetric matrices satisfying

     

Sharp uniform convexity and smoothness inequalities for trace norms

Summary We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsC _p , which are the analogs of the Lebesgue spacesL _p in non-commutative integration. The inequalities are all precise analogs of results which had been known inL _p , but were only known inC _p for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forC _p we obtain new and simple proofs of the known inequalities forL _p .Oblatum 7-VII-1993Work partially supported by US National Science Foundation grant DMS 88-07243Work partially supported by US National Science Foundation grant DMS 92-07703Work partially supported by US National Science Foundation grant PHY90-19433 A02     

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prékopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal—by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.     

Refining the Hermite-Hadamard inequalities for operator convex maps in multiple operator variables

Intensive studies aiming to extend the Hermite-Hadamard inequalities and to explore some properties and applications of these inequalities have been carried out recently. The contribution of this paper falls within this framework. We investigate here some refinements of the Hermite-Hadamard inequalities for operator convex maps involving multiple operator arguments.

     

Maximal inequalities,convexity inequality and their duality. I

Для положительных су бмартингалов А. М. Гарс иа [2] и автор [3] установили максимал ьное неравенство в случае общей функции Юнга в предположении, что до полнительная по Юнгу функция удовлет воряет так называемо му условию роста, т. е. что степень дополнительной функции конечна. Этот результат обобщает классическое нераве нство Дуба. В данной работе изуча ются подобные максим альные неравенства, но в случ ае, когда уже сама функция Юнга имеет конечную степе нь. Дается характеристика функ ций Юнга конечной степени, для которых может быть ус тановлено максимальное нераве нство.     

Sharp dilation-type inequalities with a fixed parameter of convexity

Sharp upper bounds for large and small deviations and dilation-type inequalities are considered for probability distributions satisfying convexity conditions of Brunn-Minkowski type. Bibliography: 17 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 54–78.     

The structure of approximate solutions of variational problems without convexity

In this work we study the structure of approximate solutions of variational problems with continuous integrands f:[0,∞)×Rn×Rn→R1 which belong to a complete metric space of functions. We do not impose any convexity assumption. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.     

Maximal inequalities,convexity inequality,and their duality. II

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Subdifferential characterization of approximate convexity: the lower semicontinuous case

It is known that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials.     

Estimates of approximate solutions and well-posedness for variational inequalities

The purpose of this paper is to estimate the approximate solutions for variational inequalities. In terms of estimate functions, we establish some estimates of the sizes of the approximate solutions from outside and inside respectively. By considering the behaviors of estimate functions, we give some characterizations of the well-posedness for variational inequalities. This work was partially supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China (10671135).