In this paper, taking Cauchy inequality for example, we discussed some problems on inequalities teaching in middle schools.
以经典的柯西不等式为例，从不等式的证明、推广以及它们的一些应用，对在中学数学教学中的一些问题进行讨论
Cauchy—inequality is an extremely important inequality. Ingeniously applies it or its promoted type nimbly, may cause some more difficult problem to be easily solved.
摘要柯西不等式是一个非常重要的不等式，灵活巧妙地应用它以及它的推广式，可以使一些较为困难的问题迎刃而解。
Secondly, through integrating by parts and using Cauchy inequality, thanks to a lemma of Gronwalls type, an energy inequality is got and a priori estimate is established.
接下来经过分部积分和Cauchy不等式，并利用Gronwall型引理得到一个能量不等式，对解的范数给出先验估计。
First of all, Cauchy inequality is used to obtain a basic inequality. Secondly, the functions of basis are made by Galerkin method, and the error estimates of eignevalues are obtained by Cauchy inequality. At last, the computational method of the approximate value of the eigenvalues turns out immediately, and accuracy of the-th approximate value is estimated by the n-tb approximate value.
首先利用Cauchy场不等式证明了一个基本不等式；其次采用Galerkin方法来构造适当的基函数，并利用Cauchy不等式给出了其特征值计算的误差佑计式；最后得到计算梁横向振动问题的特征值的近似值的算法，而且可以用第n次近似值来估计第n-1次的近似值的精确度。
Secondly, through integrating by parts and using Cauchy inequality, thanks to a lemma of Gronwall's type, an energy inequality is got and a priori estimate is established.
接下来经过分部积分和Cauchy不等式，并利用Gronwall型引理得到一个能量不等式，对解的范数给出先验估计。
Therefore, in order to simplify the proving process of these inequalities. Though reading a lot of relevant resource, we begin with the basic concept of math, and use an ingenious way――probabilistic method, which means that according to the main features of inequality theory, combining the basic concepts and formulas of probability, through creating one suitable probability model, giving some concrete meanings of random events or random variables, proving through probability theory, we discuss the Cauchy inequality, Class inequality, Jensen inequality, and several common inequality`s proofs.
因此，为了简化这些不等式的证明过程，通过阅读大量的相关资料，本文从数学的基本概念入手，运用了1种巧妙的方法——概率方法，即根据不等式的主要特征，结合概率论的1些基本概念和公式，通过建立1个适当的概率模型，赋以1些随机事件或随机变量的具体含义，再利用概率论的理论加以证明，讨论了柯西不等式，级数不等式，詹森不等式和几个1般不等式的证明。
In this paper, quadratic form theory, the European space inner product nature and Jensen inequality are three ways to prove cauchy inequality, and bthe relation between Cauchy inequality and higher mathematics was illustrated briefly.
本文运用二次型理论、欧式空间中内积性质和詹森不等式三种方法证明柯西不等式，并简要说明柯西不等式与高等数学之间的联系。
On the basis of the definition of matrix traces, this paper discusses their characteristics at first and then according to the norm of the F of square matrix and Cauchy-Schwarz inequality gives how to prove the zero matrix, unsimilar matrix, number cloth matrix, column matrix idempotent matrix and non-equality matrix.
根据矩阵迹的定义，首先给出了矩阵迹的性质，然后依据方阵的F—范数定义Cauchy—Schwarz不等式，给出了零矩阵，不相似矩阵，数幂矩阵，列矩阵，幂等矩阵及矩阵不等式的证法。
With the aid of a numerical method, we find the two off-resonant coupling modes may exhibit an antibunching effect, respectively, and there is a violation of the Cauchy-Schwartz inequality, which shows that the correlation between the two modes is nonclassical.
数值模拟表明，两个远共振耦合模可以呈现反聚束效应这一量子现象，贝尔不等式被破坏，表明这两个远共振耦合模之间的关联是量子关联。
Under this condition, applying Cauchy-Schwarz inequality to prove that asymptotic synchronization occurs for the coupled Lorenz systems with x-component coupling provided the coupling coefficient is sufficiently large.
并在此基础上采用Cauchy-Schwarz不等式证明当耦合强度足够大时仅有x-分量参与耦合的非恒同Lorenz格点系统的解出现渐近同步，即系统解的任意两个对应分量的差在时间趋向于无穷时是一个小的有界量。
This paper explains the deformation of Cauchyinequality and its spreading and application, introduces its algebraic proof and its geometrical significance, and also explains the extensive use of Cauchy-inequality by showing examples.
阐述了柯西不等式的变形和推广及其代数的证明。并通过例题说明柯西不等式的广泛应用。
Methods The minimun method and the Cauchy-Schwarz inequality were applied.
利用高等数学中的哥西不等式原理及求极小值方法。
Cauchy Schwarz's inequality is an applied mathematical formula with great value.
柯西不等式是应用价值非常大的数学公式。
Based on the theory of naturalist stucture, the inequality Cauchy-Schwarz a historic problem is analysed, studied and popularized. Besides the natural beauty of mathematical problem solving, thinking method of studying and discovering problems are explicitly illustrated.
以自然结构为理论依据，分析、研究及推广了历史名题Cauchy－Schwarz不等式，并简述了数学解题的自然审美观和研究问题及发现问题的思维方法。
This article elaborates that it is the effective way for the primary solution of conditional extreme value to use methods of geometrical mean, Cauchy inequality, collating principal and optics principles.
阐述了利用几何平均值、柯西不等式排序原理及光学原理是解决条件极值的初等解法的有效方法。
In 《 Advanced Mathematics 》, Cauchy inequality is important and have widespread applications and various forms.
柯西不等式是《高等代数》中一个重要而用途广泛的不等式，有多种表现形式。
In this paper, the authors give two propositions to find the conditional extremes of multivariable functions by using simple forms of Cauchy inequality.
给出了利用简单形式的柯西不等式求两类特殊多元函数条极值的两个命题。
The application widespread and flexibility of Cauchy inequality are show by the examples.
并举例说明柯西不等式在不等式证明中应用的广泛性和灵活性。
In the paper, we discusses the internal cause permeability and unity of Cauchy inequality in the different fields.
本文讨论柯西不等式在不同领域的内通性、渗透性和统一性。
Deductions of Index Number and Integral of Cauchy Inequality
Cauchy不等式的指数和积分推广