Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cauchy-schwarz-inequality]

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

0
votes
1answer
26 views

Find the maximum of $f(x, y) = \sum_{k = 1}^{n} x_k y_k$ subject to $\sum_{k=1}^{n} x_k^2 = 1$ and $\sum_{k=1}^{n} y_k^2 = 4$

Find the maximum of $f(x, y) = \sum_{k = 1}^{n} x_k y_k$ subject to $\sum_{k=1}^{n} x_k^2 = 1$ and $\sum_{k=1}^{n} y_k^2 = 4$ I just applied Cauchy Schwarz inequality to find $$-4 \leq |f(x, y)| \...
4
votes
2answers
66 views

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Let $f:[0,1]\to[1,3]$ be continuous. Prove $$1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}.$$ The left is just Cauchy's inequality with integral form, but ...
0
votes
0answers
68 views

If $K_1, …, K_{t' + 1}$ positive integers such that $K_1 + … + K_{t' + 1} = N$ prove the binomial coefficient inequality:

Consider $K_1, ..., K_{t' + 1}$ positive integers such that $K_1 + ... + K_{t' + 1} = N$ and $N$ is divisible by $t'$ and $0 < c < 1$ a constant and $t$ a positive integer such that $t < t'$. ...
0
votes
0answers
59 views

Prove that $\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$ [on hold]

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$$ I tried Holder Inequality (it's only the hint to get you ...
0
votes
3answers
56 views

Cyclic Olympiad Inequality

Given $a^2+b^2+c^2=1$ Prove $\sum_\text{cyc} \frac{1}{6ab+c^2}-\frac{1}{2+c^2}$ is nonnegative I have tried substituting 1 with $a^2+b^2+c^2$, but nothing is working. I’m trying to reduce it into a ...
0
votes
3answers
90 views

How to find $f(X)$ such that $\sum\limits_{cyc}a^2-f(X)[abc-(1-a)(1-b)(1-c)]\geqq\frac{3}{4}X^2$ for $abc=(X- a)(X- b)(X- c),0\leqq a,\,b,\,c\leqq X$?

We have $\sum\limits_{cyc}\,a^{\,2}\geqq \frac{3}{4}\,X^{\,2}\tag{HaiDangel29}$ with $abc= (\,X- a\,)(\,X- b\,)(\,X- c\,),\,0\leqq a,\,b,\,c\leqq X$. Here is a hint to get you started from above. For $...
0
votes
0answers
34 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(...
-2
votes
2answers
36 views

Proof of Cauchy–Schwarz inequality theorem [closed]

Let $U$ be unitary space. For any two vectors $a,b \in U$ is true that $|(a,b)|^2 \leq (a,a)(b,b)$
1
vote
3answers
73 views

Solve $\sqrt{3}\sin(x)+\cos(x)-2=0$

I need to solve the equation $$\sqrt{3}\sin(x)+\cos(x)-2=0$$ My try: I separated the radical then I squared and I noted $\cos(x)=t$ and I got a quadratic equation with $t=\frac{1}{2}$ To solve $\cos(...
2
votes
1answer
87 views

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$ [duplicate]

let $x,y,z,w\in R$,and such $x+y+z+w=2$.show that $$\sum_{cyc}\dfrac{x}{x^2-x+1}\le\dfrac{8}{3}$$ I have only solve when $x,y,z,w>0$, because $$\dfrac{x}{x^2-x+1}\le\dfrac{4}{3}x$$ so $$\sum_{...
0
votes
1answer
215 views

Inequality for $a,b,c>0$ $\sum_{cyc}\sqrt{\frac{a^3}{14a^2+4b^2}}\leq \sum_{cyc}\sqrt{\frac{a+b}{36}}$

A friend gives me the following result : Let $a,b,c>0$ then we have : $$\sqrt{\frac{a^3}{14a^2+4b^2}}+\sqrt{\frac{b^3}{14b^2+4c^2}}+\sqrt{\frac{c^3}{14c^2+4a^2}}\leq \sqrt{\frac{a+b}{36}}+\...
0
votes
2answers
40 views

Proof of Cauchy-Schwarz Inequality in probability form

In my university course, we were given the following proof of the Cauchy-Schwarz Inequality: My issue is with the last line, surely we get that: $$|E(XY)| \leq \sqrt{E(X^2)E(Y^2)}$$ but it is not ...
0
votes
2answers
50 views

A three variable inequality doubt , can I consider the three variables into just one variable , and show the inequality.

I was trying to prove the inequality : for a,b,c positive real numbers where $abc=1$ prove $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}}\leq 1 . $$ It is easy ...
1
vote
0answers
61 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(\...
1
vote
0answers
18 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)^...
0
votes
2answers
80 views
+50

Prove/disprove $\sum_{cyc}a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} \ge 2$ where $a$, $b$, $c > 0$ and $a^2 + b^2 + c^2 = 1$

$a$, $b$ and $c$ are positives such that $a^2 + b^2 + c^2 = 1$. Prove/disprove that $$a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} + b\sqrt{\frac{(ab + 1)(bc + 1)}{ca + 1}} + c\sqrt{\frac{(bc + 1)(ca + 1)}{...
0
votes
2answers
72 views

Tighter version of Cauchy-Schwarz?

I checked numerically that $$ \left( \sum_N p_N \dfrac{C_N}{D_N^2} \right) \left( \sum_N p_N D_N \right) \geq \left( \sum_N p_N \dfrac{C_N}{D_N} \right) $$ where $$ \sum_N p_N = 1 \;, \quad 0\leq p_N\...
1
vote
2answers
92 views

Value of $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}+\frac{1}{\sqrt{1+d^2}}$

If $$\frac{a+b+c+d}{\sqrt{(1+a^2)(1+b^2)(1+c^2)(1+d^2)}}= \frac{3\sqrt{3}}{4}$$ for $a,b,c,d>0$ Then Value of $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}+\frac{1}{...
0
votes
1answer
90 views

Solve the equation $\frac{7x^2 - x + 4}{\sqrt{3x^2 - 1} + \sqrt{x^2 - x} - x\sqrt{x^2 + 1}} = 2\sqrt 2$.

Solve the equation $\dfrac{7x^2 - x + 4}{\sqrt{3x^2 - 1} + \sqrt{x^2 - x} - x\sqrt{x^2 + 1}} = 2\sqrt 2$ over the reals. Remember when I posted a question almost identical to this one? Well, turned ...
0
votes
1answer
25 views

Proving Cauchy-Schwartz inequality without the vanish assumption of inner products

I've came accross the following question in a book that I'm studying from, about Hilbert spaces - And the answer is - What I don't understand is why implies that - ?
0
votes
1answer
45 views

Cauchy-Schwarz Inequality question

Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that \begin{align*} a^2 + b^2 + c^2 + d^2 &= 4, \\ (a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16. \end{align*} ...
1
vote
1answer
88 views

Let $n_1,\ldots,n_k$ be positive integers summing to $N$. What's an upper bound for $\sum_{i=1}^k1/\sqrt{n_i}$?

Disclaimer. Sorry, I haven't looked into this one in any detail (as I should have). I was just thinking there out-of-be an elementary principle out here (pigeon-hole, Cauchy-Schwarz, Jensen, etc.). ...
2
votes
5answers
64 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
2
votes
1answer
84 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
2
votes
1answer
105 views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

I have to prove the inequality $$ \frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}} $$ when $x^2+y^2=1$, using Cauchy-Schwarz Inequality. The RHS is equal to $\frac{12}...
0
votes
1answer
34 views

Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Under what conditions on a square matrix $A$ of size $n$ do we have $|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ? Notes The above inequalities hold for $A \in \{0, I\}$, and so by simple ...
1
vote
4answers
83 views

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

I'm at the end of an inequality proof that started out complex and I was able to simplify it to: $$\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca} \quad\text{where}\quad a, b, c ...
2
votes
3answers
28 views

What is that inequality - Cauchy-Schwarz for a single random variable?

In the following article: https://link.springer.com/content/pdf/10.1023%2FA%3A1018054314350.pdf below equation (4.1) there is a statement that: $$EZ^2 \geq (EZ)^2$$, where $Z$ is a random variable. ...
0
votes
1answer
47 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 ...
1
vote
2answers
84 views

Prove $\frac{a}{b+ c}+ \frac{b}{c+ a}+ \frac{c}{a+ b}+ \frac{63}{5}\left [ \frac{2\,c^{\,2}}{(\,a+ b\,)^{\,2}}- \frac{c}{a+ b} \right ]\geqq 0$

Given that $a,\,b,\,c> 0$, prove: $$\frac{a}{b+ c}+ \frac{b}{c+ a}+ \frac{c}{a+ b}+ \frac{63}{5}\left [ \frac{2\,c^{\,2}}{(\,a+ b\,)^{\,2}}- \frac{c}{a+ b} \right ]\geqq 0$$ I only tried Buffalo ...
0
votes
1answer
48 views

How to show following inequality using the Cauchy-Schwarz Inequality?

$A$ is $M_{n\times n}$ matrix and $x,y$ are column vector. I want to show that $||A(x-y)||_2\leq ||A||_2||x-y||_2$ with the Euclidean norm. I know that for the Cauchy-Schwarz inequality, both vectors ...
1
vote
2answers
107 views

Confusing Cauchy-Schwarz Inequality Proof

Teacher proved it like this: Very elegant (way simpler than most of the ones I find online), but I'm still not convinced -- particularly the last two steps or so, where the absolute value on the left-...
1
vote
1answer
76 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 ...
0
votes
2answers
76 views

Prove that $\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that $$\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$$ I find when the equality occurs at $a=\frac{1}{...
1
vote
3answers
50 views

Show that $f_{n}(x)=\frac{x}{1+nx^{2}}$ converges uniformly.

Show that $f_{n}(x)=\frac{x}{1+nx^{2}}$ converges uniformly. I was looking at Rudin's answer to this proof: I don't understand the part I squared in red. How do I use the Cauchy-Schwarz inequality ...
0
votes
1answer
75 views

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

$a$, $b$ 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 $...
-1
votes
1answer
86 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 ...
0
votes
0answers
28 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 ...
0
votes
1answer
55 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}^...
2
votes
2answers
79 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. $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4}$$ $$\...
-2
votes
2answers
84 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'...
0
votes
2answers
38 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{...
0
votes
6answers
91 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
vote
1answer
126 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{...
0
votes
0answers
28 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 ...
0
votes
0answers
44 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 ...
-1
votes
1answer
30 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 ...
1
vote
2answers
50 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+...
0
votes
1answer
18 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}...
0
votes
4answers
75 views

Inequalities Proof [closed]

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