运用等差（比）中项求一类三角函数式的取值范围

本文论述的三角函数式的取值范围问题 ,已有许多文章论及 ,但不外乎用纯三角法 ,方程法 ,图解法等方法 .现介绍利用等差(比 )中项将其转化为求函数最值的方法 .举例如下 :例 1　已知ｓｉｎα 2ｃｏｓβ =2 ,求 2ｓｉｎα ｃｏｓβ的取值范围 .解　据ｓｉｎα 2ｃｏｓβ =2得0≤ｓｉｎα≤ 1 ,12 ≤ｃｏｓβ≤ 1 .由ｓｉｎα 2ｃｏｓβ =2× 1知ｓｉｎα ,1 ,2ｃｏｓβ成等差数列 .设ｓｉｎα =1 -ｄ ,2ｃｏｓβ =1 ｄ ( 0≤ｄ≤1 ) ,则 2ｓｉｎα ｃｏｓβ=52 - 32 ｄ ( 0≤ｄ≤ 1 ) .∴ 2ｓｉｎα ｃｏｓβ∈ [1 ,52 ].例 2　已…     

对一道习题推广的简证

课标教材数学5（必修）第54页有题： 已知a，b是互异的正数，A是a，b的等差中项，G是a，b的正的等比中项，A与G有无确定的大小关系？     

等差-等比循环矩阵的逆矩阵求法

本文给出了等差-等此循环矩阵的逆矩阵求法,推广文[5]的结果.     

自然数的等比分拆

自然数的等比分拆李建章（陕西华阴黄河工程机械厂中７１４２０２）文［１］讨论了自然数的等差分拆，本文给出自然数分拆成等比数列之和的充要条件，从而得出分拆的一种方法。一、定义把自然数表示成自然数等比数列之和的形式，叫自然数的等比分拆；公比为１的等比分拆称...     

等差和等比中项性质的另类应用

等差中项和等比中项是数列的两个重要概念,分别有如下性质：①b是a、c的等差中项←→b=a＋c;②b是a、c的等比中项←→b2=a·c（b≠0）.在题设条件具有＂2b=a＋c＂或＂b2=a·c＂结构特征的一些非数列问题,利用上述性质来处理,新颖独特,别具一格.     

借助等差中项，妙解三角函数

对称结构是数学中一个极具美感的特征结构,对称思维是数学中一个非常优秀的思想方法.结合三角函数及其相关问题中对应要素的对称关系,合理借助等差中项的性质构建对应的等差数列,利用等差数列加以转化,实现三角函数问题的创新与求解.     

求sum from i=1 to n(f(x_i))的值的方法

有些函数值的求和问题,表面上看,与周期性、等差性、等比性无关,但事实上隐含着周期性、等差性、等比性,一旦将其周期性、等差性、等比性揭示,问题便迎刃而解.笔者就从何处揭示这些隐含的特性,从哪里入手找到撬动这些特性的支点,作一些探析,以飨读者.     

椭圆积分及其应用

介绍椭圆积分的概念和椭圆积分表的编造原理,通过实例说明椭圆积分在几何学、代数学、力学等方面的应用.     

构造等差中项解高考题

等差数列是高中重要知识点之一,在许多题目中,若能根据题目的题设条件（或隐含）的特征,构造等差中项,可使问题得到快速解决,下面对08高考题举例赏析构造等差中项求最值问题．     

自然数的等差分拆公式

自然数的等差分拆公式陕西华阴市黄河工程机械厂中学李建章文［１］给出了至少含有两个奇约数的自然数的等差分拆的一个定理，本文利用对称思想，给出所有大于８的合数的等差分拆的简易方法──真因数中项对称法．为方便，本方约定ｍ、ｎ均为自然数Ｎ的真因数（即非１和Ｎ...     

一些平均值的比较

利用关于广义平均值进行比较的已知结果去比较幂平均值、Stolarsky平均值和Heron平均值,简化了某些结果的证明,并得到了一些新的结果.     

Hardy平均及其不等式

本文研究了涉及Hardy平均的不等式问题.借助于优超理论,获得了2维Hardy平均不等式成立的充要条件及n维Hardy平均不等式的充分条件.并为建立解析不等式和高维结果提供了一些方法.     

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.     

Sub- and superadditive integral means

The integral means are special Cauchy means (see, e.g., [L. Losonczi, On the comparison of Cauchy mean values, J. Inequal. Appl. 7 (2002) 11-24]) depending on one function. The two variable integral means were (independently) defined and studied by Elezovi? and Pe?ari? [Differential and integral f-means and applications to digamma function, Math. Inequal. Appl. 3 (2000) 189-196]. The comparison problem of two integral means (under differentiability conditions) was solved by Losonczi [Comparison and subhomogeneity of integral means, Math. Inequal. Appl. 5 (2000) 609-618]. Here we completely characterize the additive, sub- and superadditive integral means of n?2 variables.     

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

均值不等式的一个推广

给出几何-算术均值不等式的一个推广,实例展示其应用,说明与约束极值原理相关的一些问题也可以通过推广的几何-算术均值不等式加以解决.     

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.     

Lagrangian mean-type mappings for which the arithmetic mean is invariant

We determine the class of all pairs of the Lagrangian means forming mean-type mappings which are invariant with respect to the arithmetic mean.     

Ando–Hiai type inequalities for multivariate operator means

ABSTRACT

We present several Ando–Hiai type inequalities for n-variable operator means for positive invertible operators. Ando–Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant.     

Generalized power means and interpolating inequalities

In this paper, we introduce a multi-parameter family of generalized power means, and use their special properties to provide a new method of interpolating inequalities. We give a different refinement of an inequality of Ky Fan as a particular application of our method.