Quod Erat Demonstrandum

2012/09/21

內接長方形

Filed under: mathematics,NSS — johnmayhk @ 2:12 下午
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1. 簡單老題

2012/03/30

WPS 不等式

Filed under: Pure Mathematics — johnmayhk @ 5:04 上午
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$a_1,a_2,\dots ,a_n,b_1,b_2, \dots ,b_n,p,q$ 為正數，其中 $\frac{1}{p}+\frac{1}{q}=1$。我們有 Hölder’s inequality

$\displaystyle \sum_{k=1}^na_kb_k \le (\displaystyle \sum_{k=1}^na_k^p)^{1/p}(\displaystyle \sum_{k=1}^nb_k^q)^{1/q}$

$a_kb_k=x_k$$b_k^q=y_k$$k=1,2,\dots ,n$

(more…)

2009/02/01

卡爾松（Carleson）不等式

Filed under: Pure Mathematics — johnmayhk @ 9:25 下午
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Carleson first inequality

Let $a_1, a_2, \dots , a_n \in \mathbb{R}$, then

$(a_1 + a_2 + \dots + a_n)^2 < \frac{\pi^2}{6}(a_1^2 + 2^2a_2^2 + \dots + n^2a_n^2)$

Carleson second inequality

Let $a_1, a_2, \dots , a_n \in \mathbb{R}$, then

$(a_1 + a_2 + \dots + a_n)^4 < \pi^2(a_1^2 + a_2^2 + \dots + a_n^2)(a_1^2 + 2^2a_2^2 + \dots + n^2a_n^2)$

2008/12/03

證明不等式的基礎招式 (Part 2)

Filed under: Pure Mathematics,Teaching — johnmayhk @ 5:12 下午
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e.g. 11 For any positive integer $n$, show that $(1 + \frac{1}{n + 1})^{n + 1} > (1 + \frac{1}{n})^n$ (more…)

2008/12/02

利用 Taylor’s theorem 證明 AM >= GM

$x_1, x_2, \dots , x_n$$n$ 個正數，命 $a = \frac{x_1 + x_2 + \dots + x_n}{n}$, $g = \sqrt[n]{x_1x_2 \dots x_n}$ (more…)

2008/11/27

證明不等式的基礎招式 (Part 1)

Filed under: Pure Mathematics,Teaching — johnmayhk @ 6:36 下午
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$A - B \ge 0$，或 (more…)

2008/11/05

講座簡報：常見的不等式及奧數解題範例

Filed under: Pure Mathematics,Report — johnmayhk @ 12:32 上午
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