Questions tagged [cauchy-schwarz-inequality]

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

49 questions
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Maximum and minimum of $\cos^2x+\sin^2y$, where $x-y=\pi/4$ and $0\leq x\leq \pi$

In the book "Calculus of several variables" by Sege Lang in page 144 the author proposes the following problema: Find the extreme values of the function $$f(x,y)=\cos^2x + \cos^2y$$ subject to the ...
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Proving that the following inequality holds in positive numbers

Assume $$S=x_1+x_2+\cdots+x_n$$ where $x_i>0$. Prove the following inequality:$$\frac{S}{S-x_1}+\frac{S}{S-x_2}+\cdot+\frac{S}{S-x_n}\ge\frac{n^2}{n-1}$$with equality iff $x_1=x_2=\cdots=x_n$. My ...
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Find $\max\,2\,x- y,\,\min\,2\,x- y$

$x^{\,2}+ y^{\,2}= e^{\,2(\,x- 2\,y\,)}$. Find $$\max\,2\,x- y \tag{and min}$$ I used this to prep for another senior student's university entrance exam in $\lceil$ diendantoanhoc.net $\rfloor$ But ...
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Let $f(x, y)=ax+by+c$. Find the largest value of $r$ such that $f(x, y)>0$ for all pairs $(x, y)$ satisfying $x^2 + y^2 < r^2$?

Let $f(x, y) = ax + by + c$, where $\{a, b, c\} \subset\mathbb R$ and $c > 0$. Find the largest value of $r$ such that $f(x, y) > 0$ for all pairs $(x, y)$ satisfying $x^2 + y^2 < r^2$. I ...
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If PQRS are four sides of a quadrilateral then find the minimum value of $(p^2+ q^2 + r^2) / s^2$ with logic

If PQRS are four sides of a quadrilateral then find the minimum value of $(p^2+ q^2 + r^2) / s^2$ with logic Please help me with this. I need this immediately.
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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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Simplifying this inequality with orthonormality relation

I am trying to simplify the following inequality \begin{equation} \langle (a_0e_0+a_1e_1)|a_0e_0 +a_1e_1|^2, \overline{e}_1\rangle \end{equation} where $a_ie_i$ are eigenvalue-eigenfunction pairs, ...
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“Cauchy-Schwartz” for symplectic forms

I would like to show some sort of "Cauchy-Schwartz" inequality for symplectic maps. i.e. given a symplectic map $\phi:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ and $u := \phi(e_1),v:=\phi(f_1)$, ...
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Inequality between matrix Schatten norms

The Schatten $p$-norm $\|M\|_p$ of a (finite-dimensional) matrix $M$ is defined as $\|M\|_p=\left(\sum_i s_i^p\right)^{1/p}$ where $s_i$ are singular values of $M$. Let $P,Q$ be positive semidefinite ...
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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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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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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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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.
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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 $f(x)<g(x) \ \forall x \in [x_1,x_2]$, ...
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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 ...
Proofs of Cauchy Inequality and Triangle Inequality for other norms on $\Bbb R^n$
Traditionally, for $\Bbb R^n$ with Euclidean inner prodct on it and $\ell^2$-norm on it, we first prove the Cauchy inequality $|x\cdot y|\leq|x|\cdot|y|$ for all $x,y\in\Bbb R^n$. Then we ...