不等式

The concept of the weighted logarithmic power mean is introduced; its relation with two-parameter mean is given; the inequality for weighted logarithmic power mean is derived; the magnitude relation among upper bounds of geometric mean and arithmetic mean of geometrically convex functions are made certain.

建立了几何凸函数的对称拟算术平均不等式，对文献[1]提出的不等式进行了推广统一；引进加权对数幂平均的概念，建立起其与双参数平均之间的关系，得到加权对数平均不等式，从而确定了几何凸函数的几何平均、算术平均的上界的大小关系；最后，提出了几何凸函数的对称拟算术平均不等式的推广问题。

Instead of Gronwall\'s inequality,the non-linear Bihari inequality is crutial in dealing with the non-Lipschitz equations.Moreover,we emphasize that the dominating functions in non-Lipschitz conditions always satisfy concavity and some non-integrability near zero.The concavity is for using Jensen\'s inequality.Because of the non-integrability near zero,we can apply the comparison theorem of ordinary differential equations together with Bihari\'s inequality to yield many results.

3作为Gronwall不等式的推广，Bihari不等式在处理具有非Lipschitz系数的方程时不可或缺；同时应当注意，非Lipschitz条件中所涉及的控制函数总是满足凹性和某种零点处的不可积性：凹性是为了利用Jensen不等式；零点处的不可积性使得Bihari不等式与常微分方程中的比较定理结合在一起从而完成很多结论的证明。

Firstly, based on the stability criterion for the nominal discrete singular time-delay system, a sufficient condition for the existence of the static output feedback controller in terms of linear matrix inequality with linear matrix equality constraint is established, which ensures that the resulting closed-loop system is regular, causal and robustly asymptotically stable, and the static output feedback controller is designed by the feasible solution of the linear matrix inequality with linear matrix equality constraint.

首先基于标称离散广义时滞系统的稳定条件，以受限线性矩阵不等式形式，得到闭环离散广义时滞系统正则、因果且渐近稳定的充分条件，同时利用受限矩阵不等式的可行解给出静态输出反馈控制器的设计方法；然后采用矩阵的正交补，把求受限线性矩阵不等式的可行解问题转化为求严格线性矩阵不等式的可行解；最后的数值实例说明了所给方法的有效性和正确性。

Based on a delay-dependent bounded real lemma, the condition under which a decentralized robust H(subscript ∞) output feedback controller exists is derived and it is available to be attributed to a solution to the problem of nonlinear matrix inequality which can be expressed in terms of a homotopy function properly selected. Then, with the homotopic iteration method and the Schur complement lemma introduced, the solution can be converted into an iterative solution to the linear matrix inequality.

基于一个时滞依赖有界实引理，将系统鲁棒分散H动态输出反馈控制器的解归结为一个非线性矩阵不等式的求解问题；选取适当的同伦函数来表示该非线性矩阵不等式，采用同伦迭代算法及Schur补引理，将求解非线性矩阵不等式转化为线性矩阵不等式的迭代求解问题。

Simple control inequation proposition is presented while in combination with the basic theory of control inequation the proofs of some known inequations are provided and some new inequations are deduced.

bstract 给出一个简单的控制不等式命题，并结合控制不等式的基本理论，用它给出若干己知不等式的证明，并推广得到了一些新不等式。

Synthesis using known facts (known conditions, important inequation, or has proved inequalities as basis, using inequalities) and relevant theorems of nature, the logical reasoning, the final step out to prove the inequality, its characteristic and the idea is "from" from "guide, known as" to "see", gradually introduced conclusion.

综合法利用已知事实(已知条件、重要不等式或已证明的不等式)作为基础，借助不等式的性质和有关定理，经过逐步的逻辑推理，最后推出所要证明的不等式，其特点和思路是"由因导果"，从"已知"看"需知"，逐步推出"结论"。

Using differential calculus knowledge to testify inequality, Calculus of higher mathematics is the core and calculus method of higher mathematics is typical of the whole mathematical method, the method is introduced for calculus ideas to solve the problem of finding an inequation, to make way for obtaining inequalities can be simple, the application of differential mid-value theorem and Taylor formula illustrate some of the inequality proof method.

运用微分学知识证明不等式；微积分是高等数学的核心，微积分思想方法是高等数学乃至整个数学的典型方法，微积分思想方法的引入为解决不等式证明的难题找到了突破，用来解不等式可使解题思路变得简单，文章应用微分中值定理及泰勒公式举例说明了部分不等式的证明方法。

In the first part, we will first deal with the strong Bonnesen-style inequality (2.1.3) for closed convex curves in the plane (the numbers of formulae and references are those of them in the context below). Bonnesen had first proved the weaker inequality (2.1.2) in [12] and several years later, he outlined in his monograph [13] various Bonnesen-style inequalities including (2.1.3), he considered, however,(2.1.3) as a direct consequence of Kritikos theorem for convex bodies in higher dimensional Euclidean spaces,. Here, we will give an independent proof of the existence for inequality (2.1.3), and by the way, give an estimate on the width of the bi-enclosing annulus of closed convex curves in the plane.

具体地讲，在第一部分中，首先讨论平面上闭凸曲线的强Bonnesen型不等式(2.1.3)(公式的编号和参考文献的编号引自后面的正文)，Bonnesen在文[12]中先证明了较弱的不等式(2.1.2)，几年以后，在他的著作[13]中，讨论了多种Bonnesen型不等式，其中包括不等式(2.1.3)，不过，他把(2.1.3)作为高维欧氏空间中凸体的Kritikos定理的直接推论，我们这里对不等式(2.1.3)给出独立的存在性证明，并且还对平面闭凸曲线的bi-enclosing环的宽度给出了一个估计。

In the first part, we will first deal with the strong Bonnesen-style inequality (2.1.3) for closed convex curves in the plane (the numbers of formulae and references are those of them in the context below). Bonnesen had first proved the weaker inequality (2.1.2) in [12] and several years later, he outlined in his monograph [13] various Bonnesen-style inequalities including (2.1.3), he considered, however,(2.1.3) as a direct consequence of Kritikos\' theorem for convex bodies in higher dimensional Euclidean spaces,.

具体地讲，在第一部分中，首先讨论平面上闭凸曲线的强Bonnesen型不等式(2.1.3)(公式的编号和参考文献的编号引自后面的正文)，Bonnesen在文[12]中先证明了较弱的不等式(2.1.2)，几年以后，在他的著作[13]中，讨论了多种Bonnesen型不等式，其中包括不等式(2.1.3)，不过，他把(2.1.3)作为高维欧氏空间中凸体的Kritikos定理的直接推论，我们这里对不等式(2.1.3)给出独立的存在性证明，并且还对平面闭凸曲线的bi-enclosing环的宽度给出了一个估计。

This thesis is devoted to the study of 〓-type inequalities for martingales,such as the maximal inequalities,the inequalities for square functions,the inequali-ties for martingale transforms,the inequalities for singular integrals.

第三章证明了关于Hardy平均算子〓的双〓不等式、双〓凸引理，并通过考虑鞅的重排不等式，建立了鞅的双〓均方函数不等式、条件均方函数不等式，给出了这类不等式成立的一个充分必要条件。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。