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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: $$\int\limits_a^bf(x)^2dx\int\limits_a^bg(x)^2dx\geq\left(\int\limits_a^bf(x)g(x)dx\right)^2$$

1673 questions
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Generalization of an inequaliy related to Cauchy-Schwarz

A classic Cauchy-Schwarz problem is to show that if $p_1, p_2 \cdots p_k$ are positive real numbers with $p_1 + p_2 \cdots +p_k=1$ then $\sum_{k=1}^n (\frac{1}{p_k} +p_k)^2 \geq n^3 +2n + \frac{1}{n}$ with equality if and only if $p_i=\frac{1}{n}$…
JoshuaZ
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Difficult Cauchy Problem

Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3 a b c$. Prove the following inequality: $$ \frac{a}{b^{2} c^{2}}+\frac{b}{c^{2} a^{2}}+\frac{c}{a^{2} b^{2}} \geq \frac{9}{a+b+c} $$ I tried using the fact that $a^2 + b^2 + c^2 = 3abc$ but I could only…
Math Olympian
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Cauchy Schwarz for 3 sequences of terms?

I have something like this: $(q^2+r^2)(s^2+t^2)(u^2+v^2)\geq(qsu+rtv)^2$ Assuming q, r, s, t, u and v are non-negative, I need to prove the inequality. Is it simple enough as being able to state(or perhaps prove) that Cauchy Schwarz can extend for 3…
progstudies213
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step in proof: Cauchy-Scharz complex inequality

Is there a smart way to see this? I tried writing it out for $n=1$ ($z=x+iy,w=u+iv$), expecting it to be simple, but I…
Sha Vuklia
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Cauchy-Schwarz inequality proof help

I can't figure out why this proof of the CS inequality starts out as shown below. I understand everything afterwards. If you expand it you get an expression that is the proof. Can someone enlighten me? $$\sum_{i=1}^n \sum_{j=1}^n\left(a_ib_j -…
Jeff Johnson
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Prove $\frac {a^{3/2}}{a+b}+\frac {b^{3/2}}{b+c}+\frac {c^{3/2}}{c+a} \ge \frac {a+b+c}{\sqrt 2}$

How does one prove $$\frac {a^{\frac{3}{2}}}{a+b}+\frac {b^{\frac{3}{2}}}{b+c}+\frac {c^{\frac{3}{2}}}{c+a} \ge \frac {a+b+c}{\sqrt 2}$$ where $a, b, c$ are positive? I tried different formats. Where could that $\sqrt2$ come from?
renmom
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Cauchy Schwarz Inequality

Let $x \in \mathbb{R}^k$. Show that if there exists a $c \geq 0$ such that $x \cdot y \leq c \left\lVert y \right\rVert$ $\hspace{1cm}$ for all $y \in \mathbb{R}^k$ then $\left\lVert x \right\rVert \leq c$ I'm not sure.
Leif
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Cauchy Schwarz: Proving this inequality

For positive reals a,b,x and y, prove that $(5a^2 + 2ab + 3b^2)(5x^2 + 2xy + 3y^2) \ge (5ax + ay + bx + 3by)^2$ My attempt: By Cauchy we can show that the LHS $\ge(5ax + 2(abxy)^{1/2} + 3by)^2$ I can then show that $ay + bx \ge 2(abxy)^{1/2}$ using…
Ram Keswani
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Check the Greatest and Smallest number

Let $V_1$$=$ $\frac{7^2\:+\:8^2\:+\:15^2+23^2}{4} -\left(\frac{7\:+\:8\:+\:15\:+\:23}{4}\right)^2$ $,$ $$$$ $V_2$$=$ $\frac{6^2\:+\:8^2\:+\:15^2+24^2}{4}-\left(\frac{6\:+\:8\:+\:15\:+\:24}{4}\right)^2$ $,$ $$$$ $V_3$$=$…
Nilabja Saha
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Schwarz inequality and bra-ket notation

It's my first time working with the bra-ket notation, and I'm still not too familiar with it. Schwarz inequality states that $||^2$$\leq$ $ $. I need to deduce that $|<\psi|AB|\psi>|^2$$\leq$ $<\psi|A^2|\psi>…
user3461126
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If $a, b, c>0$ and $a+b+c=1$, then $\left(a+\frac1b\right)\left(b+\frac1c\right)\left(c+\frac1a\right)\ge\left(\frac{10}{3}\right)^3$

Given: $a, b, c > 0$ and $a+b+c = 1$, prove that: $$\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right) \ge \left(\dfrac{10}{3}\right)^3$$ Note: this is not my own post,but instead of another user that posted here at…
Wang YeFei
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Prove that $yz+zx+xy\leq (y+z-x)^2+(z+x-y)^2+(x+y-z)^2$.

Suppose $x,y,z$ are real numbers. Prove that $yz+zx+xy\leq (y+z-x)^2+(z+x-y)^2+(x+y-z)^2$. May I ask how to use Cauchy-Schwarz Inequality to obtain $|yz+zx+xy|\leq x^2+y^2+z^2$ ? I only know $x^2+y^2+z^2\geq yz+zx+xy $. How to deal with the absolute…
sunny
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can you explain how authors used cauchy schwarz inequality for this function?

From Amir Beck book IntroductIon to nonlinear optimization p.29 We have function $f(x,y) = \frac{x+y}{x^2+y^2+1}$ "... for any $(x,y)^T \in R^2 $ , $f(x,y) = \frac{x+y}{x^2+y^2+1} <= \sqrt{2} \frac{\sqrt{x^2+y^2}}{x^2+y^2+1} <= \sqrt{2}…
Daulet Karim
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Let $x_i in $ s.t. R$\|x_i\| < 1$ for i=1,2,...,n and $x_1^2+x_2^2+...+x_n^2 = (x_1+x_2+...+x_n)^2 + 2561$. Find the least value n

Let $x_i \in \mathbb{R}$ s.t. $|x_i|<1$ for i=1, 2,..., n and $x_1^2+x_2^2+...+x_n^2 = (x_1+x_2+...+x_n)^2 + 2561$. Find the least value n so that the equation is true. My guess is to use Cauchy-Schwarz inequality so that I have…
almond_ch
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Is the Cauchy Schwarz inequality generalizable to any natural power $ p \in \mathbb{N}$?

I was wondering if we let $ p $ be a natural number, would the following hold ? $$ \left( \sum_{k=1}^{n}x_ky_k \right)^p \leq \sum_{k=1}^{n}x_k^p\sum_{k=1}^{n}y_k^p $$ I believe it should. If we reformulate this in terms of dot product, we would…
sleepydrmike
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