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Questions tagged [cauchy-schwarz-inequality]

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

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

$a$, $b$ 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
75 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
22 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
48 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
70 views

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

If $a$, $b$ anc $c$ are three positives such that $abc = 1$ then prove that $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$$ Here's what I did. $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4}$$ $$\...
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2answers
78 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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32 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
81 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 ...
1
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1answer
106 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
24 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
40 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
28 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
48 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
17 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
67 views

Inequalities Proof [closed]

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

Maximizing $\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$ for $x$, $y$, $z$ the angles of a non-acute triangle

$x$, $y$, $z$ are the angles of a triangle, one of which is not less than $\frac{\pi}{2}$. Find the maximum value of the expression $$\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$$ I ...
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4answers
80 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
37 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
47 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
55 views

$f\left ( a,\,b,\,c \right )\leqq f\left ( a,\,1,\,c \right )$ with $abc= 1$ and $a,\,b,\,c> 0$

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
54 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 ...
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2answers
45 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 ...
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1answer
16 views

A direct result from the definition of operator norm

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 $A$...
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1answer
50 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} & = &...
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1answer
58 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
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1answer
49 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 ...
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2answers
119 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_{...
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1answer
34 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}...
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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
62 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_{...
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1answer
24 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
27 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 $\mathbb{Z}$-...
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3answers
290 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
28 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 ...
2
votes
6answers
179 views

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. ...
2
votes
5answers
78 views

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 ...
1
vote
3answers
68 views

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 ...
2
votes
2answers
112 views

show this inequality $\sum\frac{x}{2+xy}\ge\frac{1}{2}$

Let $x,y,z\ge 0$ such that $$x+y^2+z^3=1.$$ Show that $$\dfrac{x}{2+xy}+\dfrac{y}{2+yz}+\dfrac{z}{2+zx}\ge\dfrac{1}{2}$$ I try do $$\sum_{cyc}\dfrac{x}{2+xy}=\sum_{cyc}\dfrac{x^2}{2x+x^2y}\ge\...
2
votes
2answers
43 views

Exercise 5.12 from Casella’s Book

Any hint for this exercise from Casella´s book: I tried with Cauchy Schwarz, Minkovsky Inequality but I am stuck. I also tried to calculate the Variance but its not clear that they are independent. ...
1
vote
0answers
43 views

Cauchy Schwarz inequality with 1 norm

Here is the argument I am making By Holder's inequality, we have for $\frac{1}{p} +\frac{1}{p^*} = 1$ $$\langle A, B\rangle \leq ||A||_p||B||_{p^*}$$ The Schatten p-norms also obey $||A||_p \geq ||...
1
vote
1answer
40 views

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that? I've managed to find only a lower bound, from $ xyz=2+x+y+z \le \frac{(x+y+z)^3}{27}...
1
vote
1answer
84 views

Showing an inequality using Cauchy-Schwarz

I managed to solve the following inequality using AM-GM: $$ \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)} \geq \frac{3}{4} $$ provided that $a,b,c >0$ and $abc=1$. However it was ...
8
votes
2answers
101 views

If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$?

It is easily seen that if $x,y\in[0,\pi/2)$ satisfy $\tan(x)\tan(y)=1$, then $$\cos(x)+\cos(y)\le\sqrt 2$$ A much more delicate fact is that if $\tan(x)\tan(y)\tan(z)=1$ (while $0\le x,y,z<\pi/2$),...
1
vote
1answer
16 views

Using a Euclidean norm to bound a $k$-tuple

This does not look too complicated, but I've been stuck here for several hours. My question is to prove that $||(h, \cdots, h)||\leq ||h||^{k}$, where $||\cdot||$ is the euclidean norm, and $(h,\cdots,...
0
votes
2answers
80 views

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Let $a, b, c>0$ s.t. $abc (a+b+c)=3$. Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+ \frac{1}{c^2+a^2+1}\leq 1$. I have no idea how to start.
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votes
3answers
81 views

Proving $\sum\sqrt{A_i^2+B_i^2} \geq \sqrt{\left(\sum A_i\right)^2+\left(\sum B_i\right)^2} $ [closed]

I know this inequality is true, but I don't know how to prove it. $$\sum_{i=1}^n\sqrt{A_i^2+B_i^2} \geq \sqrt{\left(\sum_{i=1}^nA_i\right)^2+\left(\sum_{i=1}^nB_i\right)^2} $$ Any simple equation ...
0
votes
1answer
24 views

Inequality of cyclic expression

For $a,b,c,d$ positive given that $abcd=1.$ Look at cyclic expression (i.e. rotating values in order that of a-b-c-d-a doesn't affect the result of equation) $E$ s.t. $E=\sum_{abcd}\frac{a}{da+a+1}$ ...
-1
votes
1answer
106 views

Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$. [duplicate]

Let $a, b, c, d\geq 0$ s.t. $a+b+c+d=4$. Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$. I don't know how can I deconditioned the inequality.
0
votes
0answers
49 views

Inequality using lengths of the edges of a triangle

If $a,b,c $ are the lengths of the edges of a triangle, show that: $$\frac {6 (a^2+b^2+c^2)}{a+b+c}\geq \frac {(a+b)^2}{b+c}+\frac {(b+c)^2}{a+c}+\frac {(c+a)^2}{a+b} $$ I have no idea how to start.