The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function

In the paper, by the Cauchy integral formula in the theory of complex functions, an integral representation for the reciprocal of the weighted geometric mean of many positive numbers is established. As a result, the reciprocal of the weighted geometric mean of many positive numbers is verified to be a Stieltjes function and, consequently, a (logarithmically) completely monotonic function. Finally, as applications of the integral representation, in the form of remarks, several integral formulas for a kind of improper integrals are derived, an alternative proof of the famous inequality between the weighted arithmetic and geometric means is supplied, and two explicit formulas for the large Schröder numbers are discovered.     

加权AM-GM-HM不等式链的加细

在一定条件下得到加权的算术平均、几何平均和调和平均不等式链的一种加细形式,并由此推出加权的 ｋｙ Ｆａｎ 不等式和王王不等式．此外,还给出一个连接并且分离加权算术平均和加权调和平均的单调连续函数     

Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making

In this paper, we investigate the multiple attribute decision making (MADM) problems with uncertain linguistic information. Motivated by the ideal of Bonferroni mean and geometric Bonferroni mean, we develop two aggregation techniques called the uncertain linguistic Bonferroni mean (ULBM) operator and the uncertain linguistic geometric Bonferroni mean (ULGBM) operator for aggregating the uncertain linguistic information. We study its properties and discuss its special cases. For the situations where the input arguments have different importance, we then define the uncertain linguistic weighted Bonferroni mean (ULWBM) operator and the uncertain linguistic weighted geometric Bonferroni mean (ULWGBM) operator, based on which we develop two procedures for multiple attribute decision making under the uncertain linguistic environments. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Lévy–Khintchine Representations of the Weighted Geometric Mean and the Logarithmic Mean

In the paper, the authors introduce the concept “degree of complete Bernstein function”, establish Lévy–Khintchine representations of the weighted geometric mean and the logarithmic mean by Cauchy integral formula, and obtain that the weighted geometric mean and the logarithmic mean are complete Bernstein functions of degree 0.     

三角犹豫模糊Heronian平均算子及其在多属性决策中的应用

针对属性间含有关联信息的三角犹豫模糊多属性决策问题,提出一种基于三角犹豫模糊Heronian平均算子的决策方法。首先,考虑已有Heronian平均算子仅适用输入变量为两参数的情形,进一步定义三参数加权Heronian平均(TPWHM)和三参数加权几何Heronian平均(TPWGHM)算子,证明所提算子具有还原性、幂等性、单调性和有界性等性质,并研究它们的几种特例。其次,将TPWHM和TPWGHM算子推广至三角犹豫模糊决策环境下,提出三角犹豫模糊三参数加权Heronian平均(HTFTPWHM)和三角犹豫模糊三参数加权几何Heronian平均(HTFTPWGHM)算子,并证明算子的优良性质。在此基础上,提出基于两种算子的决策分析方法,并用来解决三角犹豫模糊决策问题。最后,通过算例验证所提方法的可行性和合理性。     

加权平均值不等式推广到无穷多个数的形式

本文把有限个数的调和、几何、算术加权平均值不等式推广到无穷多个数的形式和广义积分的形式。     

犹豫模糊语言Heronian平均算子在多属性决策中的应用

针对输入变量之间的相互影响以及评价值为犹豫模糊语言信息的多属性决策问题,提出一种基于犹豫模糊语言Heronian平均算子的多属性决策方法。由于Heronian平均(HM)算子具有能够反映输入变量之间相互关联的良好特性,在犹豫模糊语言信息环境下,提出了两种新的集成算子,即犹豫模糊语言Heronian平均(HFLHM)算子和犹豫模糊语言几何Heronian平均(HFLGHM)算子,同时研究了它们的一些特性。考虑到输入变量具有不同的重要程度,还定义了犹豫模糊语言加权Heronian平均(HFLWHM)算子和犹豫模糊语言加权几何Heronian平均(HFLWGHM)算子。最后提出了基于HFLWHM算子和HFLWGHM算子的犹豫模糊语言多属性决策方法,并通过实例验证了这些算子的合理性和可行性。     

Generalized OWA Aggregation Operators

We extend the ordered weighted averaging (OWA) operator to a provide a new class of operators called the generalized OWA (GOWA) operators. These operators add to the OWA operator an additional parameter controlling the power to which the argument values are raised. We look at some special cases of these operators. One important case corresponds to the generalized mean and another special case is the ordered weighted geometric operator.     

一个Gauss型函数方程

给出了任意两个正实数的几何-调和平均值的一个积分表示式,并由此去探讨了函数方程f(ab,2ab/a+b)=f(a,b),a,b＞0其中f:R+×R+→R是此方程的一个未知函数.     

Weighted integral inequalities for the maximal geometric mean operator in martingales

Under appropriate conditions on Young's functions Φ₁ and Φ₂, we give necessary and sufficient conditions in order that weighted integral inequalities hold for the maximal geometric mean operator G in martingale Orlicz classes. When Φ₁=tp and Φ₂=tp, the inequalities revert to the ones of strong or weak (p,p)-type in martingale spaces.     

Picture模糊幂几何Heronian平均算子及其在多属性决策中的应用

针对决策信息为Picture模糊数的多属性决策问题,将经典范畴内的几何Heronian平均算子和幂几何算子结合,提出了Picture模糊幂几何Heronian平均(PFPGHM)算子与Picture模糊加权幂几何Heronian平均(PFWPGHM)算子。该类算子不仅能体现待集结数据间的关联性,而且还能反映决策过程中信息的整体性,降低了与整体信息偏差较大的待集结数据对决策结果的影响。推导其数学表达式,证明相关性质。提出了基于PFWPGHM算子的多属性决策方法,通过决策实例分析了参数p和q对决策结果的影响,并对比分析新方法与现存的决策方法,进而表明所研究方法的可行性与优点。     

凸函数的算子凸性估计及其在算子平均中的应用(英文)

本文利用Mond-Peari方法对凸函数的n个算子凸性进行了比式估计.在此基础上,得到了n个代数平均的幂与n个算子的幂的代数平均的比较,n个算子的几何平均的幂与n个算子的幂的几何平均的比较.特别地,文中还给出了代数平均和混序几何平均.     

正定自共轭四元数矩阵的均值

本文引进了两个正定自共轭四元数矩阵的算术均值，几何均值，调和均值三概念，给出了正定自共轭四元数矩阵的算术－几何－调和均值不等式，得到了正定自共轭四元数矩阵的几何均值的一个最大性质及其相关的某些性质．     

关于算术、对数、指数、几何平均不等式的概率证法

本文用概率方法证明了关于两个相异的正数的几何平均、对数平均、指数平均和算术平均的不等式关系 .     

关于Seiffert平均的一个不等式

给出了与两正数a,b的几何平均、算术平均、Seiffert平均有关的一个不等式.     

Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions

The equivalence of sequences of probability measures jointly with the extension of Skorohod's representation theorem due to Blackwell and Dubins is used to obtain strong convergence of weighted sums of random elements in a separable Banach space. Our results include most of the known work on this topic without geometric restrictions on the space. The simple technique developed gives a unified method to extend results on this topic for real random variables to Banach-valued random elements. This technique is also applied to the proof of strong convergence of some statistical functionals.     

On weighted extensions of Cauchy's means

We show that every Cauchy mean in (0,∞) can be embedded into two parameter family of weighted means. Some basic properties and examples are presented. A functional equation which appears in the problem of symmetry of these means is considered. As an application a natural extension of Stolarsky's means is obtained and a two parameter subclass of weighted power means is determined.     

On Solvability of Regular Hypoelliptic Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper the unique solvability of regular hypoelliptic equations in multianisotropic weighted functional spaces is proved by means of special integral representation of functions through a regular operator. The existence of the solutions is proved by constructing approximate solutions using multianisotropic integral operators.     

Multidimensional Jensen’s operator on a Hilbert space and applications

Motivated by a joint concavity of connections, solidarities and multidimensional weighted geometric mean, in this paper we extend an idea of convexity (concavity) to operator functions of several variables. With the help of established definitions, we introduce the so called multidimensional Jensen’s operator and study its properties. In such a way we get the lower and upper bounds for the above mentioned operator, expressed in terms of non-weighted operator of the same type. As an application, we obtain both refinements and converses for operator variants of some well-known classical inequalities. In order to obtain the refinement of Jensen’s integral inequality, we also consider an integral analogue of Jensen’s operator for functions of one variable.     

Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Pólya–Szegö type inequalities of n positive invertible operators on a Hilbert space for any integer $n?3$ in terms of the following two typical non-commutative geometric means, that is, one is the higher order weighted geometric mean due to Lawson–Lim which is an extension of the Ando–Li–Mathias geometric mean, and the other is the weighted chaotic geometric mean. Among others, the Specht ratio plays an important role in our discussion, which is the upper bound of a ratio type reverse of the weighted arithmetic–geometric mean inequality.