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Questions tagged [cauchy-schwarz-inequality]

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

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6answers
47 views

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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4answers
53 views

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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3answers
133 views

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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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\...
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2answers
298 views

Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$

Prove that if $a,b,c$ are the lengths of the edges of a given triangle, then the following inequality holds: $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + ...
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2answers
284 views

Cauchy-Schwartz inequality in a convolution

I stumbled over this Cauchy-Schwartz inequality while reading (https://link.springer.com/article/10.1007%2Fs00208-014-1046-2#Equ12 , last page before acknowledgements). Here is the situation: $$\int_{\...
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2answers
56 views

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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1answer
153 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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1answer
110 views

optimization with strict inequality of variables

Maximize $f(x_1,x_2, x_3) = x_{2}+x_{3} - (x_{2}^2+x_{3}^2)$ given $\sum_{i=1}^{3}x_{i} = 1$ and $x_{i}>0$ for $i=1,2,3$. I f I assume that $x_{i}\geq0$ for $i=1,2,3$ then the solution is $x_2 = ...
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1answer
53 views

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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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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1answer
46 views

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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1answer
125 views

Operator norm vs Square Frobenius Norm

I got this problem. Be a linear transformation $L=\mathbb{R}^n \to \mathbb{R}^m $ with Matrix $A = (a_{i,j})$. $$ ||L||_{tr} = \sum_{i=1}^{n}\sum_{j=1}^{m}(a_{i,j})^2$$ and $$||L||_{op}=sup \{|L(x)| : ...
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1answer
118 views

Square root inequality cyclic

Is it true that if $a+b+c=1$, where $a$, $b$ and $c$ are nonnegative real numbers, then $$(3-2\sqrt{2})\sum\limits_{cyc}\sqrt{ab}+2\sqrt{2}-1\geq\sum\limits_{cyc}\sqrt{(1-a)(1-b)}?$$ Edit: I was told ...
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1answer
48 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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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 ...
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1answer
43 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 ...
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1answer
56 views

Finding the best upper bound of $\int_a^b\frac{\sin^2x}x\,dx$ by hand

Q: Without using any series expansions, find, by hand, the best upper bound of $$\int_a^b\frac{\sin^2x}x\,dx.$$ Attempt: An exact expression would not be expected since we would be dealing $\text{Si}...
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1answer
50 views

Inequality with summation

If $a_i$ positive numbers and $n\ge2$ (the subscripts are taken modulo $n$), how can I prove the following inequality $n\sum\limits_{k=1}^{n} \frac{1}{(n-1)a_k+a_{k+1}}\le\sum\limits_{k=1}^{n} \frac{...
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1answer
96 views

Wild inequality

I'm trying to solve this elementary inequality, but so far no clue. Can anybody solve it? I tried am/gm inequality on both side, but yield the same result hence no inequality. Also I tried other ...
6
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0answers
303 views

Can one use properties of polynomials in order to generalize the generalized Cauchy-Schwartz inequality?

Sorry about the edits guys, I forgot to add binomial coefficients, I hope I didn't cause any needless confusion. Edit(again): I've been thinking about this a bit and perhaps I should clarify the ...
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0answers
336 views

Normed spaces where $x, y \neq 0$, $\Vert x + y \Vert = \Vert x \Vert + \Vert y \Vert $ and $\forall c > 0, \ x \neq cy$

For vectors $x, y \in \mathbb{R}^n\setminus\{0\}$, under the Euclidean norm we have that $$\left\Vert x + y \right \Vert_2 = \left \Vert x \right \Vert_2 + \left \Vert y \right \Vert_2 \iff \exists c \...
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0answers
36 views

“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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0answers
248 views

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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0answers
64 views

Proof for the bound of a complex exponential function

I am carrying out sum proof of a particular calculation and I am stuck at the following step. Let there be two functions of variable $\delta$ given by $$f(\delta) = \left|\sum_{i=1}^N\frac{e^{j\pi i(\...
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0answers
19 views

Check the Greatest and Smallest number

Let $V_1$$=$ $\frac{7^2\:+\:8^2\:+\:15^2+23^2}{4} -\left(\frac{7\:+\:8\:+\:15\:+\:23}{4}\right)^2$ $,$ $$$$ $V_2$$=$ $\frac{6^2\:+\:8^2\:+\:15^2+24^2}{4}-\left(\frac{6\:+\:8\:+\:15\:+\:24}{4}\right)^...
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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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0answers
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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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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 ||...
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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_{...
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0answers
68 views

Lagrange multiplier method on Banach spaces (in Dirac notation)

I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using ...
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0answers
118 views

Any connections between Cauchy–Schwarz inequality and continuity/boundedness?

We know the Cauchy–Schwarz inequality for inner product space such that \begin{align} |\langle u,v \rangle|^2 \le \langle u,u \rangle \langle v,v \rangle \end{align} Then if we induce the norm by $\|...
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0answers
50 views

Inequalities between different inner products defined by positive definite, symmetric matrices

Let $A, B \in \mathbb{R}^{n \times n}$ be positive definite, symmetric matrices with eigenvalues larger or equal than $1$ and let $y \in \mathbb{R}^n$ be a normalized vector, i.e. $\lVert y \rVert = 1$...
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0answers
75 views

Trigonometric inequality on a average, tight Cauchy-Schwarz and Gram's inequality

Given a function $f(x,t)$ and an average measure defined as $\langle \bullet \rangle \equiv \int_\infty dx \, g(x) f(x)$, where $g(x)$ is time-independent and normalized such that $\int g^2(x)dx=1$, I ...
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0answers
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Equality in cauchy schwartz for best estimators?

From Statistical Inference by Casella and Berger: If $W$ is an unbiased estimator of $\tau(\theta)$ then $W$ is unique. Let $W'$ be another best unbiased estimator. Let $W^* = (W + W') / 2$. ...
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0answers
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Proof of Paley-Zygmund inequality

The proof of the Paley-Zygmund inequality provided in the link just says that the Cauchy-Schwarz inequality was used but doesn't show the steps. I tried to read and understand the wiki article on ...
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0answers
40 views

Cauchy-Schwarz inversion like inequality for expectactions of comonotonic functions

Given two non constant, integrable, comonotonic functions $x_1, x_2\colon [0,\infty) \to [0,1]$, i.e., both functions are non decreasing or non increasing, I need to prove that $$\big(E[x_1(T)]+E[x_2(...
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0answers
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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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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
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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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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.
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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 $f(x)<g(x) \ \forall x \in [x_1,x_2]$, ...
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0answers
17 views

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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0answers
48 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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0answers
21 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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0answers
50 views

a tighter lower bound for weighted sum of squared norm

Let $x,y\in\mathbb{R}^d$ and $\lambda_1,\lambda_2\in\mathbb{R}_+$. We know that $\lambda_1||x||^2_2+\lambda_2||y||^2_2 \geq \frac{\lambda}{2}||x+y||_2^2$, where $\lambda=\min\{\lambda_1,\lambda_2\}$...
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0answers
31 views

How to prove $\sum_{i=1}^m \sqrt{t_{i+1}-t_{i}}E(\mathscr{N}_{i})\geq E(\mathscr{N}_{1})\sqrt{t-s}\sqrt{m}$

In Stochastic analysis lecture notes P6 by Paul Bourgade, about a total variation of Brownian motion is infinite, he writes that $$E\Big( \sum_{i=1}^m \vert B(t_{i+1})-B(t_{i})\vert \Big)\rightarrow\...
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0answers
45 views

Matrix Norm Inequalities

Given $$\Vert A x_0 - {A'} {x_0'} \Vert_2 \leq \beta_1 $$ and $$ \Vert x_0 - x'_0 \Vert_2 \leq \beta_2 $$ where $A \in \mathbb{R}^{n \times n}$, $x_0 \in \mathbb{R}^n $, can we find $\gamma_{min}, \...
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0answers
181 views

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 ...