# Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

644 questions
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### How do I find the distance from a point to a plane?

I am trying to find the distance from point $(8, 0, -6)$ and plane $x+y+z = 6$. I tried solving it but I am still getting it wrong. Can anyone help me on this? Any help I would very much appreciate. ...
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### Does $\sum\limits_{i=1}^n x_i = 1$ imply $\sum\limits_{i=1}^n x_i^2 \geq \frac{1}{n}$?

Suppose we have real numbers $x_1, ..., x_n$ which satisfy $x_1 + ... + x_n = 1$. Do we have the lower bound $x_1^2 + ... + x_n^2 \geq \frac{1}{n}$? It seems intuitive that we can minimize this by ...
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### Finding equality of inequality via Cauchy-Schwarz

Assuming $p_k > 0$, $1 \leq k \leq n$ and $p_1 + p_2 + \cdots + p_n = 1$, show that: $$\sum_{k=1}^n \left( p_k + \frac{1}{p_k} \right)^2 \geq n^3 + 2n + 1/n$$ and determine necessary and ...
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### Show for any k that $\|x\|^2\|y\|^2 -( \textbf{x} \cdot \textbf{y})^2 = \frac{1}{2} \sum_{i, j=1}^k (x_i y_j - x_j y_i)^2$

I am doing some practice tasks on Cauchy-Schwarz inequality before my university classes start and I am faced with this problem. I simply have no idea how I would go about showing this, I am ...
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### Question regarding norms of Cauchy-Schwarz inequality

I am trying to solve problems related to Cauchy-Schwarz Inequality, but I can't seem to understand why, after performing the inner product on the left side, we don't take the square root of it. ...
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### Generalization of $(a+b)^2\leq 2(a^2+b^2)$

We know that, $(a+b)^2\leq 2(a^2+b^2)$. Do we have anything similar for $$\left(\sum_{i=1}^N a_i\right)^2.$$ where $a_i\in \mathbb{R}\ \ \ \ \forall\ i\in \{1,\cdots,N\}$. For $n=3$, we get \begin{...
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### A Cauchy-Schwarz-type inequality for $\int\prod_n|f_n|$

If $X_1,X_2$ have finite second moments then Cauchy-Schwarz gives $\langle |X_1||X_2|\rangle^2 \leq \langle |X_1|^2\rangle \langle |X_2|^2\rangle$ If $(X_n)_{n=1}^N$ have their $N$th moments is it ...
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Prove that if $x_1, ..., x_{n-1}$ are positive numbers and $n \geq 2$, than the following inequality holds: $x_1^2 + x_2^3 + ... + x_{n - 1}^n + \frac{1}{x_1^2 x_2^3 ... x_{n - 1}^n} \geq n + (x_1 - ... 0answers 39 views ###$L^2$-proof of Change of measure of conditional expectation Suppose that$Z\triangleq \frac{dQ}{dP},X \in L^2_{\mathbb{P}}(\Omega;\mathcal{F})$and$\mathcal{G}$is a sub-$\sigma$-algebra of$\mathcal{F}$. How can I prove that:$E_P [ Z X| \mathcal{G}]= E_{...
If $u(x) \in C([a, b]), u(a) = 0,\; u(x) = \int_{a}^{x}u^{'}(t)dt$ then $\int_{a}^{b} |u|^{2} dx \le \frac{1}{2}(b - a)^{2}\int_{a}^{b}|u^{'}(t)|^{2}dt$ The book said it can be proved using cauchy-...
### How can we show that $\left|a+b+c\right|^p-2\left|a\right|^p\le C\left(\left|b\right|^p+\left|c\right|^p\right)$?
Let $p\ge2$. How can we show that $$\left|a+b+c\right|^p-2\left|a\right|^p\le C\left(\left|b\right|^p+\left|c\right|^p\right)\;\;\;\text{for all }a,b,c\in\mathbb R\tag1$$ for some $C\ge0$? I'm only ...