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Questions tagged [cauchy-schwarz-inequality]

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

Problems about Cauchy–Schwarz inequality.

  1. C-S inequality it's the following.

Let $a_1$, $a_2$,..., $a_n$, $b_1$, $b_2$,..., $b_n$ be real numbers. Prove that: $$(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n)^2\geq(a_1b_1+a_2b_2+...+a_nb_n)^2.$$

  1. C-S inequality in the Engel form it's the following.

Let $a_1$, $a_2$,..., $a_n$ be real numbers and $b_1$, $b_2$,..., $b_n$ be positive numbers. Prove that: $$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^2}{b_n}\geq\frac{(a_1+a_2+...+a_n)^2}{b_1+b_2+...+b_n}$$

  1. C-S inequality in the integral form.

    Let $f$ and $g$ be integrable functions on $[a,b]$. Prove that: $$\int\limits_a^bf(x)^2dx\int\limits_a^bg(x)^2dx\geq\left(\int\limits_a^bf(x)g(x)dx\right)^2$$

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We have 3 hemispheres and 3 spheres each of diameter 6 units. Stack them in such a way that the height of the tower formed is maximum

We have 3 hemispheres and 3 spheres each of diameter 6 units. Stack them in such a way that the height of the tower formed is maximum. So, I know this question seems stupid because to form a tower of maximum height we just need to stack the…
Tupac Shakur
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In whiche case do we have $\left(\sum_{i=1}^n x_iy_{i}\right)^2 = \sum_{i=1}^n (x_i)^2 \sum_{i=1}^n (y_{i})^2$?

The exercice asked me first to prove that $( \sum_i x_i y_i )^2 \leq (\sum_i x_i^2)(\sum_i y_i^2)$ ( Cauchy inequality) and i managed to prove but then it asked me in which case we have the equality ($\left(\sum_{i=1}^n x_iy_{i}\right)^2 =…
it's Rayzig
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Schwarz's inequality integration - Probability distribution

Hi I am trying to solve the problem below and somehow I do not have the right idea. A particle is located between x and x + dx with probability P(x)dx. If $\displaystyle \langle |x|^n \rangle \equiv \displaystyle \displaystyle\int_{-\infty}^{\infty}…
user120250
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Prove $(a^3+b^3+c^3)(a^2+b^2+c^2)\ge(a+b+c)(a^2b^2+b^2c^2+c^2a^2)$

Prove that $(a^3+b^3+c^3)(a^2+b^2+c^2)\ge(a+b+c)(a^2b^2+b^2c^2+c^2a^2)$, where a, b, c > 0. I did LHS-RHS, broke parentheses and cancelled terms. I think I might use $a^2+b^2+c^2\ge ab+bc+ca$. Then I only need to prove: $a^5+b^5+c^5\gt…
renmom
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Maximise $\dfrac{(ax_1 +by_1 +cz_1 +d)^2}{(a²+b²+c²)}$ under constraint $a \alpha+b \beta+c \gamma +d=0$

I'm trying to prove that the condition for distance of a point $(x_1,y_1,z_1)$ from a plane $ax+by+cz+d=0$ passing through a point $(\alpha,\beta,\gamma)$ to be maximum is $\dfrac{x_1-α}{a}=\dfrac{y_1-β}{b}=\dfrac{z_1-\gamma}{c}$ i.e. the line…
Aman Kushwaha
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extension of cauchy schwarz inequality

I Want to prove $(a_1^2+a_2^2)(b_1^2+b_2^2)(c_1^2+c_2^2)>_=(a_1b_1c_1+a_2b_2c_2)^2$ not quite sure how and not only upto 3 terms but to $n$ terms, I am not even sure if this inequality is true or not but I need this result to solve the following…
Raunit singh
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Cauchy-Schwarz inequality in $L^2$

In Cauchy-Schwarz Master Class, it states that Schwarz considered the polynomial $p(t)=\int\int_S (tf(x,y)+g(x,y))^2dxdy$ to show that if $f(x,y), g(x,y)$ are not proportional, then p(t) is strictly positive. Without resorting to analysis and…
Jia Cheng Sun
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Cauchy-Schwarz Proof Question

I'm just doing some revision on the Cauchy-Schwarz inequality for Linear Algebra using this picture from my lecture notes: The line: 'as a basic consequence of the dot product listed above, f(t) > 0 for all t as an element of the real numbers' has…
cthodrums
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Proof of Cauchy-Schwartz Inequality

Cauchy-Schwartz Inequality: I am trying to prove Question 2 of Problem 2.9: Question I have the following difficulties: 1) I am not able to understand how (1) in the figure is arrived at. 2) I am not able to understand how (2) in the figure is…
Soumee
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Cauchy-Schwarz Inequality question

Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that \begin{align*} a^2 + b^2 + c^2 + d^2 &= 4, \\ (a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16. \end{align*} I have 21 as my answer since (1,1,1,1), (0,0,0,2), and…
sumi
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Demonstration using Cauchy-Schwarz inequality

Suppose $0\leq p_j\leq 1$ for $j=1,2,3...n$, so that $p_1+...+p_n = 1$. Let's $a_j,b_j \geq 1$ so that $a_j b_j \geq1$ for $j=1,2,3...n$. Demonstrate: $1 \leq \sum^{n}_{j=1}p_ja_j \sum^{n}_{j=1}p_jb_j$ This is to be solved using the Cauchy-Schwarz…
bajotupie
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Cauchy-Schwartz inequality

Let ${\bf T} = (T_1,...,T_d) \in \mathcal{B}(F)^d$, with $\mathcal{B}(F)$ denotes the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$. It is true that $$\left(\sum_{|\alpha|=n} \|{\bf T}^\alpha…
Schüler
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Is this Cauchy-Schwarz inequality (in $\mathbb{R^n}$)?

$ \forall x, y \in \mathbb{R^n}$ this fact is true: $$\sum\limits_{i=1}^n |x_iy_i| \le \left(\sum\limits_{i=1}^nx_i^2 \right)^{\frac{1}{2}} \left(\sum\limits_{i=1}^ny_i^2 \right)^{\frac{1}{2}}$$ This inequality is very similar to C.S. but is not…
ictibones
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An application of Cauchy-Schwarz inequality

For any random $m \times n$ matrix $X$ and $m \times k$ matrix $Y$, $$ E[\| X^T Y \|] \leq E[ \|X\|^2 ]^{1/2} E[ \|Y\|^2 ]^{1/2}, $$ where $E[\| X^T Y \|]$ can be interpreted as the absolute value of inner product, $|\langle X,Y \rangle|$, and for…
dchao
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$AM$- $QM$ inequality

Show that $AM \leq QM$ is a direct consequence of the Cauchy- Schwarz inequality. I have been trying this one but I couldn't solve. Please help.
Super MaxLv4
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