Questions tagged [cauchy-schwarz-inequality]

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

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1answer
25 views

Proving Cauchy-Schwartz inequality without the vanish assumption of inner products

I've came accross the following question in a book that I'm studying from, about Hilbert spaces - And the answer is - What I don't understand is why implies that - ?
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1answer
48 views

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*} ...
1
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1answer
90 views

Let $n_1,\ldots,n_k$ be positive integers summing to $N$. What's an upper bound for $\sum_{i=1}^k1/\sqrt{n_i}$?

Disclaimer. Sorry, I haven't looked into this one in any detail (as I should have). I was just thinking there out-of-be an elementary principle out here (pigeon-hole, Cauchy-Schwarz, Jensen, etc.). ...
2
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5answers
82 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
2
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1answer
89 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
2
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1answer
105 views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

I have to prove the inequality $$ \frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}} $$ when $x^2+y^2=1$, using Cauchy-Schwarz Inequality. The RHS is equal to $\frac{12}...
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1answer
34 views

Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Under what conditions on a square matrix $A$ of size $n$ do we have $|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ? Notes The above inequalities hold for $A \in \{0, I\}$, and so by simple ...
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4answers
102 views

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

I'm at the end of an inequality proof that started out complex and I was able to simplify it to: $$\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca} \quad\text{where}\quad a, b, c ...
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3answers
30 views

What is that inequality - Cauchy-Schwarz for a single random variable?

In the following article: https://link.springer.com/content/pdf/10.1023%2FA%3A1018054314350.pdf below equation (4.1) there is a statement that: $$EZ^2 \geq (EZ)^2$$, where $Z$ is a random variable. ...
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1answer
49 views

Bounding integral of square root by square root of integral

Let $f(x)\geq 0$ be a function over $[0,\infty)$. How can I lower bound $\int_{x=0}^{u}\sqrt{f(x)}dx$ by $c \sqrt{\int_{x=0}^{u}f(x)dx}$ where $\sqrt{\int_{x=0}^{u}f(x)dx}<\infty$ and $c>0$ is ...
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2answers
149 views

A nice Nesbitt inequality from a strange inequality

Given $a,\,b,\,c> 0$$,$ prove that$:$ $$\frac{a}{b+ c}+ \frac{b}{c+ a}+ \frac{c}{a+ b}+ \frac{63}{5}\left [ \frac{2\,c^{\,2}}{(\,a+ b\,)^{\,2}}- \frac{c}{a+ b} \right ]\geqq 0$$ See$:$ $\lceil$ ...
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1answer
49 views

How to show following inequality using the Cauchy-Schwarz Inequality?

$A$ is $M_{n\times n}$ matrix and $x,y$ are column vector. I want to show that $||A(x-y)||_2\leq ||A||_2||x-y||_2$ with the Euclidean norm. I know that for the Cauchy-Schwarz inequality, both vectors ...
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2answers
135 views

Confusing Cauchy-Schwarz Inequality Proof

Teacher proved it like this: Very elegant (way simpler than most of the ones I find online), but I'm still not convinced -- particularly the last two steps or so, where the absolute value on the left-...
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1answer
93 views

Show that equality holds in the Cauchy-Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ . for x, y if and only if x and y are linearly dependent…

Could someone please help me with this proof? I have written up this but I not sure if it is a full proof, since it is an if and only if statement. Could someone read and inform me where I can go ...
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2answers
76 views

Prove that $\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that $$\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$$ I find when the equality occurs at $a=\frac{1}{...
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3answers
57 views

Show that $f_{n}(x)=\frac{x}{1+nx^{2}}$ converges uniformly.

Show that $f_{n}(x)=\frac{x}{1+nx^{2}}$ converges uniformly. I was looking at Rudin's answer to this proof: I don't understand the part I squared in red. How do I use the Cauchy-Schwarz inequality ...
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1answer
76 views

Prove that $\sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$ [duplicate]

$a$, $b$ and $c$ are positives such that $ab + bc + ca = 3abc$. Prove that $$ \sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$$ Here's what I did. My stupidity has reached a spiritual level. We have that $...
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1answer
87 views

If $a+b+c=1$ and a,b,c >0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$ [duplicate]

If $a+b+c=1$ and a,b,c>0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$. I tried with CS Engel form,homogenization but ina anyway i can't prove inequality. Can ...
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0answers
28 views

Beginner questions about applying Cauchy-Schwarz inequality correctly to RVs

Background: These are super boring questions but I'm trying to learn about CS inequality for probability... any help would be greatly appreciated. Thank you. Say $x = (1,2)$ and $y = (3,4)$ then ...
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1answer
55 views

Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

For $a_1, \ldots , a_n \in \mathbb{R}, a_1 < a_2 < \cdots <a_n$ and $a_i \ne 0$, show that $\dfrac{n}{a_1 - a_0} + \dfrac{n - 1}{a_2 - a_1} + \cdots + \dfrac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^...
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2answers
88 views

Prove that $\sum_{cyc}\frac{a}{a + b^4 + c^4} \le 1$ where $abc = 1$

If $a$, $b$ anc $c$ are three positives such that $abc = 1$ then prove that $$ \sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$$ Here's what I did. $$ \sum_{cyc}\frac{a}{a + b^4 + c^4}$$ $$\le \sum_{cyc}\...
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2answers
85 views

Prove $1/(x^5+y^2+z^2)+1/(x^2+y^5+z^2)+1/(x^2+y^2+z^5) \leq 3/(x^2+y^2+z^2)$ when $xyz \geq 1$ ($x,y,z$ are positive real numbers) [closed]

Prove $$ \frac{1}{x^5+y^2+z^2} + \frac{1}{x^2+y^5+z^2} + \frac{1}{x^2+y^2+z^5} \leq \frac{3}{x^2+y^2+z^2} ,$$ when $xyz \geq 1$ ($x,y,z$ are positive real numbers). I need this for lemma but I don'...
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2answers
44 views

Proof of Cauchy-Schwartz inequality with dot product and euclidean norm

I have some problems on understanding the proof of Cauchy-Schwartz inequality from my textbook: Given $\textbf{x,y} \in \mathbb{R} \Rightarrow \vert \textbf{x}^T \textbf{y} \vert \le \Vert \textbf{...
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6answers
92 views

Solve the equation $x^2 + 4(\sqrt{1 - x} + \sqrt{1 + x}) - 8 = 0$

Solve the equation $x^2 + 4(\sqrt{1 + x} + \sqrt{1 - x}) - 8 = 0$. Let $\sqrt{1 + x} = a$, $\sqrt{1 - x} = b$. I tried doing this. "$1 - x^2 = [\sqrt{(1 - x)(1 + x)}]^2 = (ab)^2$. The original ...
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1answer
132 views

Prove $\sum \sqrt{\frac{a^2}{6a^2+5ab+b^2}}\le \frac{\sqrt{3}}{2}$

Let $a,b,c\in R^+$ prove that the inequality $$\sqrt{\frac{a^2}{6a^2+5ab+b^2}}+\sqrt{\frac{b^2}{6b^2+5bc+c^2}}+\sqrt{\frac{c^2}{6c^2+5ca+a^2}}\le \frac{\sqrt{3}}{2}$$ My try:$$\sum\limits_{cyc} \sqrt{...
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0answers
30 views

Triangle Inequality of Tensor Products

If $$\|A - x\|_1 \le \epsilon$$ and $$\|B - y\|_1 \le \epsilon$$ where $A, B, x, y \in Herm(H_A)$, where $Herm(H_A)$ are the set of Hermitian matrices in a Hilbert space $H_A$, then can we say, by ...
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0answers
46 views

prove the following inequality using am gm hm or weirstrass etc

For $a_0,a_1,a_2,......,a_n \in R, \,\, a_0<a_1<a_2<....<a_n$ show that $$ \frac n{a_1-a_0}+\frac {n-1}{a_2-a_1}+....+\frac 1{a_n-a_{n-1}} \ge \sum_{k=1}^n \frac {k^2}{a_k} $$ i recently ...
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1answer
30 views

prove the inequality using inequalities like AM GM HM OR CAUCHY or WEIRSTRASS ETC.

The inequality to be proven is $$ 2^n \gt 1 + n\cdot \sqrt{2^{n-1}} for\ all\ n>2 $$ using any inequalities like am gm hm cauchy schwarz tchebychev etc I recently studied inequalities came across ...
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2answers
51 views

Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place

Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place: $$ \frac{x}{y^3(1+y^2x)} + \frac{y}{z^3(1+z^2y) } + \frac{z}{x^3(1+...
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1answer
18 views

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}...
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4answers
75 views

Inequalities Proof [closed]

if $x+y+z ≤ 3$ is it necessarily true that $$1/x + 1/y + 1/z ≥3?$$ Thanks!
3
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4answers
88 views

minimum value of $2\cos \alpha\sin \beta+3\sin \alpha\sin \beta+4\cos \beta$

Let $\alpha,\beta$ be real numbers ; find the minimum value of $2\cos \alpha\sin \beta+3\sin \alpha\sin \beta+4\cos \beta$ What I tried : $\bigg|4\cos \beta+(2\cos \alpha+3\sin \alpha)\sin \beta\...
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2answers
39 views

$a,b,c\in\Bbb R^+, x,y,z\in \Bbb R, $ show that $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c} \ge \frac{(x+y+z)^2}{a+b+c}$

$a,b,c\in\Bbb R^+, x,y,z\in \Bbb R, $ show that $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c} \ge \frac{(x+y+z)^2}{a+b+c}$ (use Cauchy–Schwarz inequality) I have trouble finding the two vectors. Is it $(...
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2answers
50 views

$x,y\in\Bbb R,$ find the maxima of $\frac{x+2y+3}{\sqrt{x^2+y^2+1}}$

I want to use Cauchy–Schwarz inequality, I sqared $\frac{x+2y+3}{\sqrt{x^2+y^2+1}}$ and got $\frac{x^2+4y^2+9}{x^2+y^2+1}$, not sure if I am doing fine
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1answer
134 views

Find $k=constant$ such that $f(a,\,b,\,c\,)=\frac{3a+2b}{\sqrt{5a^2-ab+b^2}}+\frac{3b+2c}{\sqrt{5b^2-bc+c^2}}\leqq f(a,k\,a+\sqrt[3]{abc}-k\,c,c\,)$

Give $3$ positve numbers $a,\,b,\,c$ such that $abc= 1$ , prove: $$f\left ( a,\,b,\,c \right )= \frac{3\,a+ 2\,b}{\sqrt{5\,a^{\,2}- ab+ b^{\,2}}}+ \frac{3\,b+ 2\,c}{\sqrt{5\,b^{\,2}- bc+ c^{\,2}}}\...
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4answers
69 views

Some Cauchy-Schwarz Inequalities [closed]

I am trying to learn how to deal with inequalities to prepare for a Math Olympiad and right now I am working on Cauchy-Schwarz. However, I am not that good at seeing the relationships and I don't have ...
3
votes
2answers
51 views

Cauchy-Schwarz Inequality troubles

I have to prove the following inequality using the Cauchy-Schwarz inequality: $$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge2$$ where a, b, c and d are positive real numbers. But ...
2
votes
1answer
35 views

A direct result from the definition of operator norm: $\|Av\|\leq \|A\|_{op} \|v\|$

Although Wikipedia says this result comes from the definition of Operator Norm directly, I am not quite sure how to understand it: Let $\|\cdot\|$ denote Euclidean norm. Given a $n\times n$ matrix $A$...
0
votes
1answer
54 views

Find minimum of sum of product of sequences

Let $a_{i}, b_{i}, c_{i},\ d_{i}$ be non-negative sequences of length $k$ such that $$ \begin{matrix} \sum_{k}a_{i} & = & nk \\ \sum_{k}b_{i} & = & nk\\ \sum_{k}c_{i} & = &...
1
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1answer
100 views

Bordered Hessian matrix to find a minimum of the function

I was trying to find the global minimum for the function $$(a + b) z + (a + c) y + (b + c) x $$ subject to the following constraint: $$(xy + xz + yz)(ab + bc + ac)=1.$$ By Lagrange multipliers I found ...
2
votes
1answer
51 views

Prove that in every triangle the inequality $a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $ takes place

Prove that in every triangle the inequality $$a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $$ takes place, with the usual notations ($a,b,c$ lengths of sides, $r_a, r_b, r_c$ radii of coresponding ...
1
vote
2answers
130 views

$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$ with $abc=1$

Let $a,b,c>0$ such that $abc=1$ then we have : $$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$$ My try : The original inequality is equivalent to : $$a(b^b+1)(c^c+1)+b(a^a+1)(...
1
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0answers
30 views

Find the minimum of a energy conservation equation.

I have an equation for two particle collisions (the equation is just energy-momentum conservation): $$k_{A,x}+k_{B,x} = p_{1,x}+p_{2,x}$$ $$k_{A,y}+k_{B,y} = p_{1,y}+p_{2,y}$$ $$k_{A,z}+k_{B,z} = p_{...
0
votes
1answer
40 views

Does Cauchy-Schwarz hold for: $ \langle\textbf{u},\textbf{v}\rangle \;\leq ||\textbf{u}|| \cdot ||\textbf{v}|| $

I am wondering whether the Cauchy-Schwarz inequality does hold when absolute value is not considered for the LHS. Let me explain: In standard Cauchy-Schwarz we have: $| \langle \textbf{u},\textbf{v}...
1
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2answers
54 views

3-variable symmetric inequality

Given $a,b,c>0$ satisfying $a^2+b^2+c^2=3$. Prove that $$2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)+3(a+b+c)\geq 15.$$ I've tried to use the inequality $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}...
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0answers
78 views

Cauchy-Schwarz Master Class Exercise 1.13

This is the question from Michael Stelle's book, exercise 1.13: Show that if $\{a_{jk} : 1\leq j \leq m, 1 \leq k \leq n\}$ is an array of real numbers then one has $$m \sum_{j=1}^m \left( \sum_{...
0
votes
1answer
45 views

Schwarz-Pick on a disc of radius R > 0…

Let $R, C > 0$. Let f be a holomorphic function defined on $D(0, R)$ and such that f is bounded above by $C$. Prove that $$|f'(z)| \leq \frac{R}{C}\cdot \frac{C^2 - |f(z)|^2}{R^2 - |z|^2}.$$ I am ...
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2answers
80 views

Cauchy-Schwarz inequality for $L^2$-norm on periodic functions space

I have proven something that is definitely not true (Lemma 2), which is why I am intersted where I err. Definition Let $C(\mathbb{R}/\mathbb{Z},\mathbb{C})$ be the set of all continuous $\mathbb{Z}$-...
4
votes
3answers
295 views

Find the smallest value of $f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$ on the interval $(0,\pi/2)$

There's a function defined as: $$f(x) := \left({1\over9}+{32\over \sin(x)}\right)\left({1\over32}+{9\over \cos(x)}\right)$$ In interval $$\left(0,\frac{\pi}{2}\right)$$ Find the smallest value (...
2
votes
1answer
37 views

Prove $l^2$ norm obeys the triangle inequality

I'm trying to work through Exercise 3 from this blog post, which is essentially a proof of the validity of the $l^2$ norm: Exercise 3: Let $(\mathcal{V},\left<\cdot,\cdot\right>)$ be an inner ...