一个经典不等式的再探究

不等式的证明历来是高考数学命题的热点和重点.给出一个经典的不等式,并对不等式作进一步的探究,开拓学生的视野,训练学生的思维.     

一个不等式的上下界再探

文[1]证明了如下不等式: 若非负实数x,y满足x y=1,λ≥1,则√x/λ y √x/λ x≤2/√2λ 1(1).     

一个不等式猜想的解决

文[1]末提出了四个不等式猜想,其中的猜想1笔者已给出了一个肯定性证明和推广[2]．最近,笔者发现猜想2也是成立的．     

一个猜想不等式的证明

文[1]介绍了如下不等:若xi>0(i=1,2,3),且∑3 i=1 xi=1.则1/(1+x21)+1/(1+x22)+1/(1+x23)≤27/10.     

一个不等式猜想的证明

文献[1]给出了如下一个不等式： 若a，b，x，y∈R^＋，则     

一个猜想不等式的推广

文[1]提出如下猜想：若a,b,c为满足a＋b＋c=1的正数,则     

一个猜想不等式的否定

贵刊2013年第2期姜坤崇先生《一个猜想不等式及其部分解决》一文给出了一个猜想：     

一个不等式猜想引出的问题

在文献[1]中，李韵教师提出了如下猜想：设a，b，c∈R＋且a＋b＋c=1，n∈N＋，     

一个积型不等式猜想的证明

文[1]提出了如下一个积型不等式猜想: 设ai＞0(i=1,2,…,n),n≥3,∑ni=1ai=1,k∈N*,则有     

一类偏微分方程特征值的上界估计

考虑一类偏微分方程特征值的上界估计,利用分部积分、Rayleigh定理和不等式估计等方法,获得了用前n个特征值来估计第n＋1个特征值的上界的不等式,其估计系数与区域的度量无关,这个结果在力学和物理学中有着广泛的应用。     

线性交簇超图边数的上界

H是线性交簇超图，｜E∩F｜：1（∨E、F∈H），记s=s（H）：min｜E｜，A：｜E∈H：｜E｜：s｜．若｜A｜〈s^2＋1，则m（H）≤A（[H]2）＋1；若｜A｜≥s^2＋1，则当s≤2时，m（H）≤△（[H]2）＋1；当s≥3时，m（H）≤△（[H]2）-2s．     

Upper bound analysis for estimation of the influence of seepage on tunnel face stability in layered soils

Tunnel face stability is important for safe tunneling and the protection of the surrounding environment. Upper bound analysis is a widely applied method to investigate tunnel face stability. In this paper, a tunnel face collapse of Guangzhou metro line 3 is presented. Accordingly, seepage is considered in the upper bound solutions for face stability in layered soils. Steady-state seepage is reached in the first 1200 s of each drilling step. In the crossed layer, the seepage flow is horizontal toward the tunnel face, whereas in the cover layer, the seepage vertically percolates into the crossed layer. By considering the seepage forces on the tunnel face and on the soil particles, the upper bound solution for the support pressure needed for face stability in layered soil with seepage is obtained. Under saturated conditions, the support pressure is influenced by the variation of the depth ratio due to the seepage effect. Moreover, the support pressure depends linearly on the groundwater level. This study provides estimations of the support pressure for face stability in tunnel design.     

Upper bound solution of supporting pressure for a shallow square tunnel based on the Hoek-Brown failure criterion

To analyze the stability of a shallow square tunnel, a new curved failure mechanism, representing the mechanical characteristics and collapsing form of this type of tunnel, is constructed. Based on the upper bound theorem of limit analysis and the Hoek-Brown nonlinear failure criterion, the supporting pressure derived from the virtual work rate equation is regarded as an objective function to achieve optimal calculation. By employing variational calculation to optimize the objective function, an upper bound solution for the supporting pressure and the collapsing block shape of a shallow square tunnel are obtained. To evaluate the validity of the failure mechanism proposed in this paper, the solutions computed by the curved failure mechanism are compared with the results calculated by the linear multiple blocks failure mechanism when the Hoek-Brown nonlinear failure criterion is converted into the Mohr-Coulomb linear criterion. The influences of rock mass parameters on the supporting pressure and collapsing block shape are discussed.     

一个不等式的证明及应用

给出一个条件不等式，并用于解几道国内外数学竞赛题。     

一个不等式赛题的引申与推广

题　设ａ、ｂ、ｃ为正数 ,且ａ ｂ ｃ=1。证明 :(1 ａ) (1 ｂ) (1 ｃ)≥ 8(1 -ａ) (1 -ｂ) (1 -ｃ)①(第 1 7届 (1 991年 )全俄数学奥林匹克十一年级题 3 )本文给出上述不等式的一个推论及一个一般性的推广。推论 1　设ａ、ｂ、ｃ为正数 ,且ａ ｂ ｃ=1 ,则(ａ ｂ) (ｂ ｃ) (ｃ ａ)≤ 82 7 ②证明 :　由不等式①得(ａ ｂ) (ｂ ｃ) (ｃ ａ)≤ 18(1 ａ) (1 ｂ) (1 ｃ)≤ 18[(1 ａ) (1 ｂ) (1 ｃ)3 ]3 =82 7。将不等式①的条件中“三个正数的和为 1”改为“ｎ个正数的和为 1”得到不等式①的如下推广。推广　设ａ1,ａ2 ,… ,ａ…     

一个不等式成立的充分条件

利用Hahn-Banach定理，给出了在线性赋泛空间中，当M包含X,x0∈X／M,f∈X,f（M）=0时，不等式｜f（x0）｜≤‖f‖ρ（x0,M）成立的一个充分条件。     

关于矩阵秩的一个不等式的注记

本文得到了关于矩阵秩的不等式r(A B)≤r(A) r(B)的更为精确的推广形式。     

Vicente Goncalves不等式的一个注记

In this paper, a new proof of the inequality of Vicente Gonqalves was given. The method may be powerful to deal with some problems of polynomials. As an application, a new inequality for the coefficients of a polynomial and its roots was derived.     

一个不等式的推广和应用

对一个不等式进行了研究 ,推广了这个不等式 ,并讨论其应用。