Inequalities - Minkowski's inequality

§ \S Suppose that r r is finite and not equal to 11. Then

¯(a)+¯(b)+...+¯(l)>¯(a+b+...+l)

\mathcal{\bar{A}}(a) + \mathcal{\bar{A}}(b) + ... + \mathcal{\bar{A}}(l) > \mathcal{\bar{A}}(a+b+ ... + l) holds with r>1 r>1; and

¯(a)+¯(b)+...+¯(l)<¯(a+b+...+l)

\mathcal{\bar{A}}(a) + \mathcal{\bar{A}}(b) + ... + \mathcal{\bar{A}}(l) holds with r<1 r. The inequality holds unless (a),(b),...,(l) (a), (b), ..., (l) are proportional, or r≤0 r\le 0 and aν=bν=...=lν=0 a_\nu = b_\nu = ... = l_\nu=0 for some a ν \nu.

The main result remains true when r=+∞ r=+\infty or r=−∞ r=-\infty, except that the conditions for equality require a restatement.

§ \S If r r is finite and not equal to 00 or 1 1, then

(∑(a+b+...+l)r)1/r<(∑ar)1/r+...+(∑lr)1/r

\left(\sum(a+b+ ... + l)^r\right)^{1/r} holds with r>1 r>1, and

(∑(a+b+...+l)r)1/r>(∑ar)1/r+...+(∑lr)1/r

\left(\sum(a+b+ ... + l)^r\right)^{1/r} > \left(\sum a^r\right)^{1/r} + ... +\left(\sum l^r\right)^{1/r} holds with r<1 r.

§ \S If r r is positive and not equal to 11, then

∑(a+b+...+l)r>∑ar+...+∑lr

\sum(a+b+ ... + l)^r > \sum a^r + ... +\sum l^r holds with r>1 r>1, and

∑(a+b+...+l)r<∑ar+...+∑lr

\sum(a+b+ ... + l)^r holds with r<1 r.

In a nutshell, what is usually required in practice is as follows:
If r>0 r>0 then,

(∑(a+b+...+l)r)R<(∑ar)R+...+(∑lr)R

\left(\sum(a+b+ ... + l)^r\right)^{R} where R=1 R=1 if 0<r≤1 0 and R=1r R=\frac{1}{r} if r>1 r>1.

The following inequality is often useful for the prupose of determining an upper bound for ∑ak \sum a^k.

§ \S Suppose that k>1 k>1, that k′ k' is conjugate to k k, and that B>0B>0. Then a necessary and sufficient condition that ∑ak≤A \sum a^k \le A is that ∑ab≤A1/kB1/k′ \sum ab \le A^{1/k}B^{1/{k'}} for all b b for which ∑bk′≤B\sum b^{k'}\le B.

The geometrical interpretations are illustrated as follows. When k=2 k=2, A=l2 A=l^2, and B=1 B=1 holds, we take rectangular coordinats. The theorem asserts that, if the length of the projection of a vector along an arbitrary direction does not exceed l l, the length of the vector does not exceed ll.

Inequalities - Minkowski's inequality相关推荐

泛函分析1-线性空间
文章目录 Preface 1 Linear Spaces 1.1 linear space Examples 1.2 Subsets of a linear space Notation 1.3 Su ...
上海交大课程MA430-偏微分方程续论(索伯列夫空间)之总结(Sobolev Space)
我们所用的是C.L.Evans "Partial Differential Equations" $\def\dashint{\mathop{\mathchoice{\,\rlap ...
中心极限定理_High Dimensional Probability(1) 中心极限定理
1. 大数定理与中心极限定理关于大数定理有一个很常见的描述,就是掷一枚硬币很多次,最终出现的正面次数和反面次数各占比 ,即: 当n足够大时,某一事件事件出现的频率将几乎接近于其发生的概率.大数定理描 ...
9.2 向量范数的三大不等式
文章目录柯西-施瓦茨不等式赫尔德不等式闵可夫斯基不等式我这里要讲的三大不等式不是三种范数比较大小的三大不等式.而是非常经典的,学习线性代数必须掌握的三大不等式:柯西-施瓦茨不等式.赫尔德不 ...
度量空间（metric space）
参考文章:(GTM135)Advanced Linear Algebra 度量空间定义度量空间(metric space)是二元组(M,d)(M,d)(M,d),其中MMM是非空集合,度量(met ...
Boole‘s，Doob‘s inequality，中心极限定理Central Limit Theorem，Kolmogorov extension theorem, Lebesgue‘s domin
1. Boole's inequality In probability theory, Boole's inequality, also known as the union bound, says ...
Cauchy-Schwarz Inequality
Cauchy-Schwarz Inequality Keyword : Cauchy–Schwarz inequality Minkowski inequality Young's inequalit ...
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) (/mɪŋˈkɔːfski, -ˈkɒf-/[1]) is a co ...
INEQUALITY BOOKS
来源:这里 Bất Đẳng Thức Luôn Có Một Sức Cuốn Hút Kinh Khủng, Một Số tài Liệu và Sách Bổ ích Cho Việc Học ...

Inequalities - Minkowski's inequality

Inequalities - Minkowski's inequality相关推荐

最新文章

热门文章