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Questions tagged [cauchy-schwarz-inequality]

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

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Cauchy-Schwarz Complex Variable Inequality

For part (ii), it is easy for me to show that the said inequality holds for $z^3 - 2$, as taking the reciprocal of both sides and flipping the inequality gives me a nice inequality which can be used ...
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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\...
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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.
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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 ...
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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>...
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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{(...
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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{...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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). ...
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1answer
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 - ...
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It's $a\cdot b + b\cdot c + c\cdot a\leq 12$? [closed]

Let $a, b, c >0$ with $a^3+b^3+c^3=8$. It's $a\cdot b + b\cdot c + c\cdot a\leq 12$? I want to prove another inequality and it's sufficiently to prove this inequality.
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$L^2$-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_{...
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1answer
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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-...
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1answer
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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 ...
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Maximize product of a vector with two vectors

Say we're in a (complex) Hilbert space $H$ where we're given some two elements $a,b\in H$ of norm 1. The question is to find an element $\psi\in H$ (of norm 1) that maximizes $|\langle \psi,a\rangle \...
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Prove $x_1x_2…x_n \ge (n-1)^n$ [closed]

Let $x_1x_2…x_n$ be positive real numbers such that $ $$\tfrac{1}{1+x_1}$$ + $$\tfrac{1}{1+x_2}$$ +… + \tfrac{1}{1+x_n}=1 $ Prove that $x_1x_2…x_n \ge (n-1)^n$ Please can someone help with this ...
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Cauchy-Schwarz inequality to prove $A$ is spd

Let $\vec v$ be a non-zero vector in $\Bbb R^n$ such that $\|\vec v\| = 1$ and let $A = I −β\vec v\vec v^T$, with $β > 0$. (a) Show that if $β ≤ 1$, then $A$ is spd. Hint: use Cauchy-Schwarz ...
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How to solve this problem using Cauchy-Schwarz inequality

How to solve this problem? This is one of the problems in my semester examination: For $x,y\in\mathbb{R}^+,x^2+y^2=1$, find the maximum value of $M=\sqrt{x}+\sqrt{2y}$. I know it can be solved ...
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Find the minimum value of $\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$

Given that $0\lt x\lt 2$ and $0\lt y\lt 2$ then find the minimum value of $$\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$$ My try: On factorisation we need minimum value of $$\...
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Contradiction Observation to Cauchy Schwarz inequality for inequalty

$f:[0,1]\to \mathbb R$ , be continuous function then prove that $$\int_0^1f^2(x)dx\geq \biggl(\int_0^1|f(x)| \biggr) ^2$$ I tried this for $x^2$ For that above is true But I checked following ...
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Proof of an inequality: is it correct?

Let $x_{1},\cdots, x_{n}>-1$ be real numbers such that $\sum{x_{i}}=n$. Prove that: $$\sum_{i=1}^{n}{\frac{1}{x_{i}+1}}\geq \sum_{i=1}^{n}{\frac{x_{i}}{x_{i}^{2}+1}}$$ My proof: By AM-HM and $\...
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Revisit “Inequality between Frobenius and nuclear norm”

I am reading the following question: Inequality between Frobenius and nuclear norm I know $$\|X\|_* = \sum_{i=1}^r \sigma_i(X)$$and $$\|X\|_F = \bigg(\sum_{i=1}^r \sigma^2_i(X)\bigg)^{1/2}$$ I try ...
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Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$

Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$ If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ ...
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Show $(\sum_{k=1}^{n} |u_k|^2)^{\frac12}\leq \sum_{k=1}^{n}|u_k| \leq n\cdot max\{|u_k|:1\leq k \leq n\} $

My text book has used the following inequalities without any proof: Let $X=R^n$, then $$(\sum_{k=1}^{n} |u_k|^2)^{\frac12}\leq \sum_{k=1}^{n}|u_k| \leq n\cdot max\{|u_k|:1\leq k \leq n\} \leq n\...
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applying Cauchy-Schwarz

I am trying to work through a homework set, and the one of the problems states the following inequity: $|\int \prod_{j=1}^{2^n} f_{j}d\mu| \leq \prod_{j=1}^{2^n} (\int |f_{j}|{ ^2}^n d\mu)^{1/{2^n}}$....
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How to apply the Cauchy-Schwartz inequality on the following inner product?

Reading a research article, I came across the following statement: The following function (where $x_i$ and $p_j$ are two vectors in $\mathbb{R}^n$ and $\mu_{i,j}$ is a constant) - it doesn't matter ...
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How to draw inference about unconditional universe from a conditional one?

I was trying to work out a proof for the Schwarz inequality $$(\mathbb{E}[XY])^2 <= \mathbb{E}[X^2]\mathbb{E}[Y^2]$$ where X and Y are random variables. So, I started with the above expression ...
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1answer
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$L^1$ inequality between ordered real numbers implies $L^2$ norm inequality

Let $x_1,\ldots,x_n$ be real numbers such that $|x_1|\geq \ldots\geq |x_n|$ and $\displaystyle \sum_{i=k+1}^n |x_i| \leq \alpha \sum_{i=1}^k |x_i|$ where $1\leq k \leq n-1$ and $\alpha >0$. ...
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1answer
31 views

Is the following operator a compact operator?

I have to choose whether the following operator $$f\in L^2(\mathbb{R})\mapsto\int_{\mathbb{R}}f(x) e^{-x^2} dx$$ is a compact operator. I have tried to use the Cauchy Schwarz inequality $$\langle ...
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2answers
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Proof for Cauchy-Schwarz inequality for Trace [closed]

Cauchy-Schwarz inequality applied to Trace of two products $\mathbf{Tr}(A'B)$ has the form $$ \mathbf{Tr}(A'B) \leq \sqrt{\mathbf{Tr}(A'A)} \sqrt{\mathbf{Tr}(B'B)} $$ I saw many places where people ...
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1answer
35 views

Question concerning an application of Cauchy-Schwarz

Specifically, the question is as follows: Prove that for every integrable real-valued $f:\mathbb{R}\rightarrow\mathbb{R}$, $$\left(\int_1^ef(x)dx\right)^2\leq\int_1^ex(f(x))^2dx.$$ I'm really ...
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1answer
27 views

Proving $q:\mathbb{R}^n \to \mathbb{R} \text{ with } q(x):= x^TAx$ totally differentiable

Let $A \in \mathbb{R^n}$ be a real $n \times n$ matrix. How can I prove that the function $$q:\mathbb{R}^n \to \mathbb{R} \text{ with } q(x):= x^TAx$$ is totally differentiable on $R^n$ and find ...
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0answers
29 views

Remark 16 in Chapter 2 from Brezis - Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

Let $E,F$ be two Banach Spaces and $A : D(A) \subset E \to F$ be a linear unbounded operator which is densely defined. Now, we would like to define $A^{*}$ as the adjoint of $A$. Let $A^{*} : D(A^{*}) ...
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2answers
44 views

Maximum length perimeter of a box whose diagonal is 10 unit long.

Suppose the diagonal of a three-dimentional box has length $10$, what is its maximum perimeter length? Here is my solution: Let the three edges of the box adjacent to a vertex be labelled $a,b,c$. ...
9
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1answer
147 views

What kind of “geometric” regularity $f'^2$ gives on $f$

When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on. Today I got the following exercise : Let $f \in \mathcal{C}^1(\mathbb{R},\...
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2answers
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Finding maximum value of squared sum

Given k positive integers such that $x_1+x_2+...+x_k=n$, find the maximum possible value of $x_1^2+x_2^2+...+x_k^2$. I know we can find its minimum value using Cauchy-Schwarz inequality, but is ...
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0answers
18 views

Prove K is a convex body and find the polar of K (K°)

Say $K$ = { x ∈ $R^3$ : $\frac{x_1^2}{a_1}$ + $\frac{x_2^2}{a_2}$ + $\frac{x_3^2}{a_3}$ $\leq$ 1 }. Show that K is a convex body. I've managed to show that K is centrally symmetric and closed. How can ...
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1answer
40 views

Inequality involving a kind of Harmonic mean

While revising the Harmonic mean, I came across this inequality which I haven't figured out how to solve, but I think it should be the application of some known inequality. I would be very grateful if ...
4
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1answer
55 views

If $f$ is continuous and $3\geq f(x)\geq 1$ for all $x\in[0,1]$, show the integral inequality.

If $f$ is continuous and $3\geq f(x)\geq 1$ for all $ x\in [0,1]$, show that $$ 1 \leq \int_0^1f(x)dx\int_0^1\Bigg(\frac{1}{f(x)}\Bigg)dx \leq \frac{4}{3}. $$ Thanks!
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4answers
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Prove using squared number property

$$ If \sum_{i=1}^{10} x_i=10 $$ Prove that $$ \sum_{i=1}^{10} x_i^2\ge 10 $$