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Questions tagged [cauchy-schwarz-inequality]

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

3
votes
3answers
156 views

prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$

Let $x,y,z\ge 0$,and $x+y+z=2$, show that $$\sqrt{1-xy}+\sqrt{1-yz}+\sqrt{1-xz}\ge 2$$ Mt try: $$\Longleftrightarrow 3-(xy+yz+zx)+2\sum_\text{cyc}\sqrt{(1-xy)(1-yz)}\ge 4$$ or $$\sum_\text{cyc}\sqrt{(...
3
votes
3answers
69 views

Prove $x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$

Prove $$x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$$ I have a solution, and I hope to see more nicer ones, thanks for your interest! We have $$(\,x+ y+ z\,)^{\,2}+ (\,-\...
2
votes
5answers
79 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 $|u \cdot v| \leq ||u|| \ ||v||$ ...
1
vote
1answer
33 views

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

$a$, $b$ and $c$ are non-negatives such that $a + b + c = 3$. Prove that $$\large 3 \le a\sqrt{b^3 + 1} + b\sqrt{c^3 + 1} + c\sqrt{a^3 + 1} \le \frac{ab^2 + bc^2 + ca^2}{2} + 3$$ This problem is ...
0
votes
2answers
53 views

Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z) $

Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z) $ Professor says Cauchy-Schwarz theory should be used.
2
votes
2answers
88 views

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

Using the Cauchy-Schwarz Inequality, we have $$ \begin{align} 1 &=\left(\int_n^{n+1}1\,\mathrm{d}x\right)^2\\ &\le\left(\int_n^{n+1}x\,\mathrm{d}x\right)\left(\int_n^{n+1}\frac1x\,\mathrm{d}x\...
0
votes
1answer
52 views

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2} )\ge 4$, preferably with SOS. My approach: $\sum_{\mathrm{cyc}} \frac{a^2(b^2+c^2)+3abc(b+c)}{3bc(b^2+c^2)} \ge 4$. Let $f(a,b,c)=\...
2
votes
2answers
86 views

Prove that $\sum_{cyc}\frac{a}{a + b^4 + c^4} \le 1$ where $abc = 1$

If $a$, $b$ anc $c$ are three positives such that $abc = 1$ then prove that $$ \sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$$ Here's what I did. $$ \sum_{cyc}\frac{a}{a + b^4 + c^4}$$ $$\le \sum_{cyc}\...
0
votes
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 ...
1
vote
3answers
56 views

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

$x$, $y$ and $z$ are positives such that $xyz = \dfrac{1}{2}$. Prove that $$ \frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ...
2
votes
2answers
76 views

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

$a$, $b$ and $c$ are positives such that $3a + 4b + 5c = 12$. Calculate the maximum value of $$\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$$ I want to know if there are ...
0
votes
3answers
32 views

Let n be a positive integer such that

$\sin {\frac {π}{2n}}+\cos {\frac {π}{2n}}=\frac{\sqrt n}{2}$. Then A) $6\le n\le8$, B) $4\lt n\le8$, C) $4\le n\le 8$, D) $4 \lt n\lt8$, I couldn't get started in solving it. That's why ...
2
votes
9answers
168 views

Minimize this real function on $\mathbb{R}^{2}$ without calculus?

When it comes to minimizing a differentiable real function, calculus comes into play immediately. If $f: (x,y) \mapsto (x+y-1)^{2} + (x+2y-3)^{2} + (x+3y-6)^{2}$ on $\mathbb{R}^{2}$, and if one is ...
2
votes
1answer
56 views

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

$x$, $y$ and $z$ are positives such that $xy + yz + zx = 1$. Prove that $$\large \frac{1}{x^2 + 1} + \frac{1}{y^2 + 1} + \frac{1}{z^2 + 1} \ge \frac{2}{3}\left(\frac{x}{\sqrt{x^2 + 1}} + \frac{y}{\...
3
votes
4answers
80 views

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

$a$, $b$ and $c$ are positives such that $\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{c^2 + a^2} = 1$. Calculate the minimum value of $$\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}$$ I ...
1
vote
5answers
91 views

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Prove for all positive a,b,c $$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$$ My Try I tried taking common denominator of the expression, $\frac{a^2b+ab^2+b^2c+c^2b+ac^2+a^2c}{abc}$ How to ...
0
votes
2answers
68 views

Find the minimum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}}$ where $a, b, c > 0$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 3$.

$a$, $b$ and $c$ are positives such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \le 3$. Calculate the maximum value of $$\large \frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}} + \frac{1}{\sqrt{b^2 - bc + 3c^...
5
votes
5answers
111 views

Find Maxima and Minima of $f( \theta) = a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta$

Show that, whatever the value of $\theta$, the expression $$a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta\ $$ Lies between $$\dfrac{a+c}{2} \pm \dfrac 12\sqrt{ b^2 + (a-c)^2} $$ My ...
-3
votes
2answers
83 views

Olympiad level question on sequence and series [closed]

Example 16. If $x_1,x_2,\dots,x_n$ are $n$ non-zero real numbers such that $$\left(x_1^2+x_2^2+\cdots+x_{n-1}^2\right)\left(x_2^2+\cdots+x_n^2\right)\leq \left(x_1x_2+x_2x_3+\cdots+x_{n-1}x_n\right)^2$...
-1
votes
3answers
131 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 ...
4
votes
2answers
187 views

Show that $a+b+c+\sqrt {\frac {a^2+b^2+c^2}{3}}\le4$,

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2+abc=4.$ Show that $$a+b+c+\sqrt {\frac {a^2+b^2+c^2}{3}}\le4.$$
1
vote
3answers
88 views

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

Problem number 1: The problem is that $x, y, z$ are proper fractions, and each one of them is greater than zero. Given $x + y + z = 2$, prove $$\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$$ I have ...
2
votes
2answers
143 views

A nice Nesbitt inequality from a strange inequality

Given $a,\,b,\,c> 0$$,$ prove that$:$ $$\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$$ See$:$ $\lceil$ ...
0
votes
1answer
98 views

Holder unsuccessful here!

For $x,\,y,\,z> 0$, prove that $$\frac{y}{\sqrt{2\,z(\,x+ y\,)}}+ \frac{z}{\sqrt{2\,x(\,y+ z\,)}}+ \frac{x}{\sqrt{2\,y(\,z+ x\,)}}\geqq \frac{3}{2}$$ I tried $\lceil$ HOLDER!inequality $\rfloor$ ...
1
vote
1answer
130 views

Find $k=constant$ such that $f(a,\,b,\,c\,)=\frac{3a+2b}{\sqrt{5a^2-ab+b^2}}+\frac{3b+2c}{\sqrt{5b^2-bc+c^2}}\leqq f(a,k\,a+\sqrt[3]{abc}-k\,c,c\,)$

Give $3$ positve numbers $a,\,b,\,c$ such that $abc= 1$ , prove: $$f\left ( a,\,b,\,c \right )= \frac{3\,a+ 2\,b}{\sqrt{5\,a^{\,2}- ab+ b^{\,2}}}+ \frac{3\,b+ 2\,c}{\sqrt{5\,b^{\,2}- bc+ c^{\,2}}}\...
-1
votes
1answer
100 views

Prove $\left \{ \sum\limits_{cyc}\,\frac{(\,b+ c\,)^{\,2}}{a^{\,a}+ 1} \right \}\leqq 8$ [on hold]

Let $a,\,b,\,c,\,d$ be non-negative numbers. Prove that $$\left \{ \sum\limits_{cyc}\,\frac{(\,b+ c\,)^{\,2}}{a^{\,a}+ 1} \right \}\leqq 8$$ for $\sum\limits_{cyc}\,a^{\,2}= 4$. I found it $\lceil$ ...
1
vote
2answers
70 views

Prove $2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$ [closed]

For $a,\,b,\,c\geqq 0$ and $b\equiv {\rm mid}\,\{\,a,\,b,\,c\,\}$. Prove $$2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$$ Inspried from $\lceil$ Prove $ \sum_{cyc}a^3- \sum_{...
0
votes
0answers
23 views

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 ...
0
votes
1answer
25 views

Minimizing a summation

How can I solve this exercise? $min \quad {\sum_{i=1}^{n}{\frac{c_i}{x_i}}} $ $s.t. \quad \sum_{i=1}^n{a_ix_i}=b $ $x_i \geq 0 $ $a_i,b_i,c_i >0$ My attempts. I think I should calculate the ...
0
votes
3answers
113 views

Find $f(\,X\,)$ such that $\sum_{cyc}a^2-f(X)\left\{\prod_{sym}a-\prod_{sym}(1-a)\right\}\geqq\frac{3}{4}X^2$

Find $f(\,X\,)$ such that $$\left \{ \sum\limits_{cyc}\,a^{\,2} \right \}- f(\,X\,)\,\left \{ \prod\limits_{sym}\,a- \prod\limits_{sym}\,(\,1- a\,) \right \}\geqq \dfrac{3}{4}\,X^{\,2}$$ We have $\...
3
votes
2answers
92 views

Find minimum value of $P=17x^2+17y^2+16xy$

Let $x,y$ be positive real numbers such that $4x^2+4y^2+17xy+5x+5y\ge 1$. Find minimum value of $$P=17x^2+17y^2+16xy$$ My idea: I see that $x=y=\frac{\sqrt 2 -1}{5}\rightarrow P=2(3-2\sqrt 2)$ So I ...
0
votes
1answer
42 views

about Lagrange multipliers

For the real numbers $a, b, c$ and $d$ such that: $a+b+c+d=4$ and $a^2+b^2+c^2+d^2=s \ge \frac {28}{3}$, I have to find the maximum value of the product $abcd$ in terms of $s.$ We may use the ...
2
votes
1answer
60 views

Proving an inequality given conditions.

Let real numbers $x_1, x_2, x_3, x_4, x_5, x_6$ satisfy $x_1+x_2+x_3+x_4+x_5+x_6=0, $ and $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=6.$ Prove $x_1x_2x_3x_4x_5x_6\leq\frac{1}{2}.$ I am trying to figure out ...
2
votes
1answer
34 views

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

Exercise : Assume that $T$ satisfies the equation $T_t(t,z) = aT_{zz}(t,z)$ for $t>0, z \in (0,1)$ and $a > 0$ a constant. Moreover, suppose that $T(0,z) = T_0(z)$ for $z \in [0,1]$, where $...
0
votes
2answers
95 views

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
1answer
30 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
77 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
1answer
227 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
3answers
63 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
0answers
38 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
78 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
89 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
2answers
44 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
53 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
62 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(\...
0
votes
2answers
76 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\...
0
votes
1answer
76 views

Showing that $x+y+z \le 2\left(\frac {x^2}{y+z}+ \frac {y^2}{x+z}+ \frac {z^2}{x+y}\right)$ for positive $x, y, z$?

Please, even more than the solution I would like to understand how get better at solving inequalities. Currently my only method is to just blindly try different manipulations to see if they work. ...
0
votes
2answers
61 views

Find the maximize of $\sum_{cyc}\frac{1}{x^2+y^2+1}$

Let $x>0$, $y>0$ and $z>0$ such that $xy+yz+xz=3$. Find a maximize of $$P=\frac{1}{x^2+y^2+1}+\frac{1}{y^2+z^2+1}+\frac{1}{z^2+x^2+1}$$ We need to prove $P\le 1$ with $x=y=z=1$ We have: $$\...
1
vote
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)^...