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Questions tagged [cauchy-schwarz-inequality]

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

2
votes
6answers
186 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
85 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
97 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
130 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
49 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
84 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
44 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
85 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
103 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
17 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
81 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.
-1
votes
3answers
84 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
26 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
109 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
51 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.
3
votes
1answer
79 views

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Let $a,b,c,d>0$ satisfying $${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$$ Prove that $abcd\geq 81$ I've tried to apply arithmetic geometric mean inequality or Cauchy-Schwartz ...
5
votes
4answers
816 views

determining if sequence has upper bound

I am somewhat stuck in my calculations when determining if sequence has an upper bound. The sequence $$x_n = \frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n-1}+\frac{1}{2n}$$ Is equal to $$\frac{1}{n}(\...
0
votes
1answer
33 views

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 ...
0
votes
2answers
41 views

minimum value of $l$

Minimum positive real number $l$ for which $7\sqrt{a}+17\sqrt{b}+l\sqrt{c}\geq 2019.$ given that $a+b+c=1$ and $a,b,c>0$ what i try cauchy Inequality $$(7^2+17^2+l^2)(a+b+c)\geq \bigg(7\sqrt{...
2
votes
2answers
65 views

How to prove $\frac{(x+y)(y+z)(z+x)}{4xyz}≥\frac{x+z}{y+z}+\frac{y+z}{x+z}$ for $x,y,z>0$?

Prove that for $x,y,z$ positive numbers: $$ \frac{(x+y)(y+z)(z+x)}{4xyz}≥\frac{x+z}{y+z}+\frac{y+z}{x+z} $$ I tried to apply MA-MG inequality: $x+y≥2\sqrt{xy}$ and the others and multiply them but ...
2
votes
3answers
66 views

maximum and minimum of $x^2+y^4$ for real $x,y$

If $y^2(y^2-6)+x^2-8x+24=0$ then maximum and minimum value of $x^2+y^4$ is what i try $y^4-6y^2+9+x^2-8x+16=1$ $(x-4)^2+(y^2-3)^2=1\cdots (1)$ How i find maximum and minimum of $x^2+y^4$ from $(1)...
0
votes
3answers
66 views

Extending Cauchy-Schwarz to any $p \in (1,\infty)$

Given $n$ non-negative real number $\left\lbrace a_i\right\rbrace_{i=1}^n$, we know that using Cauchy-Schwarz, one can prove that $$ \sum\limits_{i=1}^n \sqrt{a_i} \leq \sqrt{n} \cdot \sqrt{\sum\...
0
votes
1answer
55 views

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. ...
0
votes
2answers
52 views

Inequality with $ax^2+bx+c$

Let $a,b,c,x,y > 0$ reals prove that: $$(ax^2+bx+c)(ay^2-by+c) \geq (4ac-b^2)xy$$ What I have done is this: $$ax^2+bx+c=a \left (x+\frac{b}{2a} \right)^2+\frac{4ac-b^2}{4a}$$ $$ay^2-by+c=a \left (...
0
votes
0answers
26 views

Constructing an integral based on Cauchy-Shwarz.

I don't know exactly how to phrase my question, but I am attempting to construct an integral that satisfies the following: $\int_{x_1}^{x_2}f(x)g(x)dx=1$ if $f(x)<g(x) \ \forall x \in [x_1,x_2]$, ...
2
votes
2answers
173 views

Cauchy Schwarz - Finding minimum of a function

For $x, y , z$ in real numbers, If $2x+y+z=5$, then what is the min value of $x^2 + y^2 + z^2 - 2x + 4y + 6$. This is a weekend brain teaser in the 2nd week of Calc 3.
1
vote
2answers
69 views

Trig Integral Inequality

Show that if $f$ is Riemann integrable on $[a,b]$, then $$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2\le(b-a)\int_{a}^{b}f^2(x)\ dx.$$ I know I need to use ...
0
votes
3answers
53 views

Is this inequality true? $u^2+v^2+s^2+t^2\geq (u+v)(s+t)$

Is this inequality true in $\mathbb{R}$? $$u^2+v^2+s^2+t^2\geq (u+v)(s+t)$$ I don't know if this is a well-known result. If you have a counterexample or a relevant reference I would appreciate it.
0
votes
1answer
71 views

Does Cauchy-Schwarz imply $|x^Ty| \leq \|x\|_p\|y\|_p$ for any $p \geq 1$?

Given $x,y \in \mathbb{R}^n$, the Cauchy Schwarz inequality states, $|x^Ty| \leq \|x\|_2\|y\|_2$ And for non-Euclidean (norms other than $l_2$), we have, $|x^Ty| \leq \|x\|_p\|y\|_q$ where $\|\cdot\...
0
votes
3answers
117 views

Cauchy Schwartz Inequality Question: $(a^2+b^2)^3=c^2+d^2 \implies \frac{a^3}{c}+\frac{b^3}{d}\geq 1$

If $(a^2+b^2)^3=c^2+d^2$, prove that $\frac{a^3}{c}+\frac{b^3}{d}\geq 1$. Please help.
0
votes
1answer
40 views

Cauchy-Schwarz inequality for points on unit sphere?

I have the following problem (I added a photo of the problem): For a set of points $x \in \pi_k$ with $x \in R^d$ and on unit sphere we compute: $m_j = \frac{1}{n_j}\sum_{x \in \pi_k} x $ and ...
3
votes
3answers
65 views

Prove $\frac{1}{3}(a+b+c)^2\leq a^2 + b^2 + c^2 + 2(a-b+1).$

Prove that for $a>1$,$b>1$ and $c>1$ where $a,b,c\in \mathbb{R}$ $$\frac{1}{3}(a+b+c)^2\leq a^2 + b^2 + c^2 + 2(a-b+1).$$ My attempt: it is not so clear why is $a>1$, $b>1$ and $c>...
0
votes
1answer
39 views

Inequality triangle Radon substitutions

I have this inequality: $$\sum \frac {a^3}{p-a}\geq 8(2R-r)^2$$ I have tried using Radon substitutions and I get this: $$\sum \frac{(y+x)^3}{x}\geq 8(2R-r)^2$$ I know from Holder that : $$\sum \frac{(...
5
votes
3answers
113 views

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{...
1
vote
1answer
81 views

Schwarz inequality in linear algebra and probability theory

Linear algebra states Schwarz inequality as $$\lvert\mathbf x^\mathrm T\mathbf y\rvert\le\lVert\mathbf x\rVert\lVert\mathbf y\rVert\tag 1$$ However, probability theory states it as $$(\mathbf E[XY])^2\...
0
votes
2answers
42 views

Proving Inequalities Involving Summations and Sq. Roots

How can I prove that $\sum_{i=1}^{n} |a_i| \leq \sqrt{n} \sqrt{\sum_{i=1}^{n} a_{i}^{2}}$ considering that $ a_{1}, a_{2}, a_{3}, ... , a_{n} $ are real numbers? This exercise was presented in a ...
0
votes
1answer
66 views

Prove the inequality $\sum_{cyc} {{a+abc} \over {1+ab+abcd}} \ge {{10} \over {3}}$ with Cauchy-Schwarz [closed]

Problem: If $abcde = 1$, $a, b, c, d, e > 0$, $a, b, c, d, e \in \Bbb R$, prove that $\sum_{cyc} {{a+abc} \over {1+ab+abcd}} \ge {{10} \over {3}}$ First I proceeded with Cauchy-Schwartz ...
2
votes
2answers
51 views

Inequality in 3 variables with a constraint condition

To prove : a(b+c)/bc + b(c+a)/ca + c(a+b)/ab > 2/(ab+bc+ca) where a+b+c=1 and a,b,c are positive real numbers Here's my way : Add 1 to each summand in the LHS and subtract 3 (=1+1+1) and after some ...
0
votes
1answer
199 views

Does the matrix norm inequality or the Cauchy-Schwarz inequality hold for L2,1 norms

I read here https://statweb.stanford.edu/~souravc/Lecture32.pdf that Cauchy-Schwarz inequality holds for the Hilbert-Schmit or Frobenius norms. I wanted to know if the same holds for other norms too ...
4
votes
3answers
59 views

Maximizing $f$ in $\mathbb{R}^3$

Find the domain and the maximum value that the function $$f(x,y,z)=\frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}$$ may attain in its domain. I have found the domain of the function to be $\mathbb{R^3\...
3
votes
1answer
47 views

Interchanging limit and integral.

Suppose $(X,\mu)$ is a probability space, $W\in L^1(X)$, $V\in L^\infty(X)$, and $V_n\to V$ in $L^2(X)$ (in my situation $V_n$ is the partial Fourier sum and so the $L^2(X)$ convergence is automatic). ...
2
votes
1answer
212 views

Prove $ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $

Show that if $a,b,c > 0$, such that $ab + bc + ca = 1$, then the following inequality holds: $$ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $$ What I ...
3
votes
1answer
39 views

How to show for any orthogonal vector set and any matrix $\sum_u \|Au\|_2^2 \leq \|A\|_F^2$?

The following paper on page 16, in line 17 Online Principal Component Analysis says for any orthogonal vector set and any matrix $\sum_u \|Au\|_2^2 \leq \|A\|_F^2$ is true. However, if we let $u_i$'s ...
1
vote
1answer
41 views

How to show $\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$?

Let $x,y \in \mathbb{R}^n$. How can I show the following $$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$ The above has been used by the authors of the following paper on page 8, in first line Online ...
2
votes
1answer
63 views

Proving a Cauchy-Schwarz-like inequality

For real numbers $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ with $$x_1,y_1>0,\ x_1^2>x_2^2+\ldots+x_n^2,\ y_1^2>y_2^2+\ldots+y_n^2,$$ show that $$x_1y_1-x_2y_2-\ldots-x_ny_n \geq \sqrt{(x_1^2-...
1
vote
1answer
43 views

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 ...
0
votes
1answer
97 views

Prove $x_1^2 + x_2^3 + … + x_{n - 1}^n + \frac{1}{x_1^2 x_2^3 … x_{n - 1}^n} \geq n + (x_1 - 1)^2 + 2(x_2 - 1)^2 + … + (n - 1)(x_{n - 1} - 1)^2$

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 - ...
1
vote
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_{...
2
votes
1answer
67 views

How to prove the following by Cauchy-Schwarz? [duplicate]

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-...
1
vote
1answer
52 views

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 ...