对数核Toader平均的Schur凸性和Schur几何凸性

为了研究对数核Toader平均L_r(a,b)在R_(++)~2上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1/2时,L_r(a,b)在R_(++)~2上是Schur凹函数;当r≥0时,L_r(a,b)在R_(++)~2上是Schur几何凸函数;当r≤0时,L_r(a,b)在R_(++)~2上是Schur几何凹函数,最后,依据L_r(a,b)的Schur凸性和Schur几何凸性建立了新的不等式.     

线性核Toader平均的Schur凸性和Schur几何凸性

为了研究线性核Toader平均Mr(a,b)在R_(++)²上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1时,M_r(a,b)在R_(++)2上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1时,M_r(a,b)在R_(++)²上是Schur凸函数;当r≤1时,Mr(a,b)在R_(++)2上是Schur凸函数;当r≤1时,Mr(a,b)在R_(++)²上是Schur凹函数;当r≥1/2时,M_r(a,b)在R_(++)2上是Schur凹函数;当r≥1/2时,M_r(a,b)在R_(++)²上是Schur几何凸函数.最后,依据M_r(a,b)的Schur凸性和Schur几何凸性建立了新的不等式.     

N 元指数和对数平均的凸性及几何凸性

讨论了n 元指数平均和对数平均的凸性、S - 凸性、几何凸性及S - 几何凸性, 证明了:(1) n 元指数平均是S - 凹的和S - 几何凸的; (2) n 元第一对数平均是S - 凹的; (3) n 元第二对数平均是凹的和几何凸的. 最后提出了二个悬而未决的问题.     

关于拉格朗日中值定理与中间值的唯一性

拉格朗日中值定理是: 如果(i)函数f(x)在闭区间[a,b]连续,(ii)f(x)在开区间(a,b)可微,那么在(a,b)内至少存在一点ξ,使得     

有关凸函数的一个定理及其应用

引理1(1)若f(x)为区间[a,b]上的凸函数,对于x1、x2、x3∈[a.b],且满足x1≤x2≤x3,则总有f(x2)-f(x1)/x2-x1≤f(x3)-f(x2)/x3-x2.①(2)若f(x)为区间[a,b]上的凹函数,对于x1、x2、x3∈[a,b],且满足x1≤x2≤x3,则总有f(x2)-f(x1)/x2-x1≥f(x3)-f(x2)/x3-x2.     

弹性函数的若干应用

列举了弹性函数在数学分析、经济分析和数值分析三个方面的应用.在数学分析方面利用弹性函数分别给出了函数超(次)加性和几何凸(凹)性的判别定理,在经济分析和数值分析方面利用弹性函数进行了简单的弹性分析和误差分析.     

连续函数的l凸性

在研究函数的性态时,笔者发现如下定义的l凸函数,它反映了函数中普遍存在的凸偏移现象.定义:设f(x)是定义在实数集D上的实值函数,常数l∈R,若对 xk∈M( D),pk≥ 0,k=1,2,…,n, (n∈N,n≥2),∑nk=1pk=1,都有f(∑ni=1pixi+l)≤∑ni=1pif(xi)则称f(x)为M上的l凸函数;当-f(x)为l凸函数时,称f(x)为M上的l凹函数.下面给出连续函数具有l凸性的两个判定定理:定理 1:设f(x)是定义在 [a,a+2l] (l>0)上的连续的增函数,则f(x)是 [a,a+l]上的l凹函数,也是[a+l,a+2l]上的(-l)凸函数.证明:设xi∈[a,a+l] (i=1,2,…,n),x1≤x2≤…≤xn,则xi+l∈[a+l,a…     

凸函数的Hadamard不等式的若干推广

本文获得两个定理 ,它们均是不等式f a +b2 1b -a∫baf (x) dx f (a) +f (b)2(其中 f是 [a,b]上的连续凸函数 )的推广 .     

一个加细的反向Jensen不等式

文 [1 ]在函数的凸性理论中 ,给出了一个重要的结论 :设 f ( x)、p( x)为 I上的可积函数 ,而 m≤ f ( x)≤ M,p( x)≥ 0 ,∫Ip( x) dx >0 ,则随连续函数Φ( t) ( m≤ t≤ M)之为下凸或上凸而相应地有Φ∫Ip( x) f ( x) dx∫Ip( x) dx≤或≥∫Ip( x) f ( x) dx∫Ip( x) dx(即 Jensen不等式 )　　为证明其反向不等式 ,引入以下记号 ,并引入严格凸函数的一个几何性质。记 I =[a,b];∫I=∫ba;A( f ( x) ) =∫Ip( x) f ( x) dx∫Ip( x) dx为 f ( x)的加权平均 ,p( x)≥ 0 ,∫Ip( x) dx >0 ,x∈ I。设Φ( x) >0 ,Φ″( x) >0 ,x∈ I,则Φ( …     

学习微商与不定积分(續)

二、微商的应用我們有了微商的知識,可以利用它研究函数的一些主要性貭,例如利用微商可以确定函数的单調性(增加或减小)、函数的极大极小,进而研究最大最小問题,做为几何上的应用,可以研究函数曲綫的凸凹性、作函数图形等等。利用微商研究这些問題时,要时时刻刻不忘一个函数的微商在几何上表示函数曲綫的切綫斜率这一几何意义,并密切联想函数图形,就容易理解上述諸問題中利用微商的思路,最后利用微商解决定未定式的問题。 1.微分学中的重要定理研究上述諸問題以前,先介紹一下两个重要定理。它将刻划函数在整个区間上的变化与微商概念的局部性之间的联系。中值定理若函数f(x)在閉区間[a,b]上連續,在开区間(a,b)內可微,則在(a,b)內必有一点ξ,使 (f(b)-f(a))/(b-a)=f＇(ξ)。 (3) 这个定理从图形上看是很明显的,設函数f(x)所表示的曲綫如图2。A和B的纵坐标各为f(a)和f(b),因此     

Duality for quasi-concave programs with explicit constraints

The idea of duality is now well established in the theory of concave programming. The basis of this duality is the concave conjugate transform. This has been exemplified in the development of generalised geometric programming. Much of the current research in duality theory is focused on relaxing the requirement of concavity. Here we develop a duality theory for mathematical programs with a quasi concave objective function and explicit quasi concave constraints. Generalisations of the concave conjugate transform are introduced which pair quasi concave functions as the concave conjugate transform does for concave functions. Optimality conditions are derived relating the primal quasi concave program to its dual. This duality theory was motivated by and has implications in certain problems of mathematical economics. An application to economics is given.     

N元指数和对数平均的凸性及几何凸性

讨论了n元指数平均和对数平均的凸性、S-凸性、几何凸性及S-几何凸性，证明了：（1）n元指数平均是S-凹的和S-几何凸的；（2）n元第一对数平均是S-凹的；（3）n元第二对数平均是凹的和几何凸的．最后提出了二个悬而未决的问题．     

Notes on L-/M-convex functions and the separation theorems

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000     

A geometric framework for nonconvex optimization duality using augmented lagrangian functions

We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal and dual problems. Using the primal function of a constrained optimization problem, we apply our results both in the analysis of duality schemes constructed using augmented Lagrangian functions, and in establishing necessary and sufficient conditions for the convergence of penalty methods.     

The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.     

一类对称函数的Schur凸性及其应用

对x=(x_1,…,x_n)∈[0,1)~n∪(1,+∞o)~n,定义对称函数■其中r∈N,i_1,i_2,…,i_n为非负整数.研究了F_n(x,r)的Schur凸性、Schur乘性凸性和Schur调和凸性.作为应用,用控制理论建立了一些不等式,特别地,给出了高维空间的一些新的几何不等式.     

On Duality for a Class of Quasiconcave Multiplicative Programs

Multiplicative programs are a difficult class of nonconvex programs that have received increasing attention because of their many applications. However, given their nonconvex nature, few theoretical results are available. In this paper, we study a particular case of these programs which involves the maximization of a quasiconcave function over a linear constraint set. Using results from conjugate function theory and generalized geometric programming, we derive a complete duality theory. The results are further specialized to linear multiplicative programming.     

Geometrically concave univariate distributions

In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.     

Trigonometric Convex Underestimator for the Base Functions in Fourier Space

A three-parameter (a, b, x_s) convex underestimator of the functional form (x) = -a sin[k(x-x_s)] + b for the function f(x) = sin(x+s), x [x_L, x_U], is presented. The proposed method is deterministic and guarantees the existence of at least one convex underestimator of this functional form. We show that, at small k, the method approaches an asymptotic solution. We show that the maximum separation distance of the underestimator from the minimum of the function grows linearly with the domain size. The method can be applied to trigonometric polynomial functions of arbitrary dimensionality and arbitrary degree. We illustrate the features of the new trigonometric underestimator with numerical examples.Support from the National Science Foundation and the National Institutes of Health Grant R01 GM52032 is gratefully acknowledged.     

Some properties of a class of symmetric functions and its applications

For , the symmetric functions are defined by where , and are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.