Skip to main content

Questions tagged [rearrangement-inequality]

Inequalities related to the rearrangement inequality or Chebyshev inequality

Filter by
Sorted by
Tagged with
3 votes
1 answer
92 views

Prove that $ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c$

Let $ a, b, c > 1 $. Prove that $ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c. $ My approach: I denoted $ \...
math.enthusiast9's user avatar
5 votes
1 answer
260 views

Showing $\frac1{ab+4}+\frac1{ac+4}+\frac1{ad+4}+\frac1{bc+4}+\frac1{bd+4}+\frac1{cd+4}\geq\frac65$, for positive $a,b,c,d$ with $ab+bc+cd+da=4$

Let us take $a\geq b\geq c\geq d>0$ such that $ab+bc+cd+da=4$. Show that $$\frac{1}{ab+4}+\frac{1}{ac+4}+\frac{1}{ad+4}+\frac{1}{bc+4}+\frac{1}{bd+4}+\frac{1}{cd+4}\geq\frac{6}{5}$$ $a+b+c+d \ge 4$...
Math learner's user avatar
1 vote
2 answers
93 views

Use of rearrangement inequality

Let (x,y,z) be real numbers, each greater than (1). Prove that $$\dfrac{x+1}{y+1}+\dfrac{y+1}{z+1}+\dfrac{z+1}{x+1} \leq \dfrac{x-1}{y-1}+\dfrac{y-1}{z-1}+\dfrac{z-1}{x-1}$$ I found a solution here, ...
zaemon_23's user avatar
  • 589
0 votes
1 answer
133 views

$x, y, z > 0, xyz = 1 => xy^2 + yz^2 + zx^2 \geq x+y+z$ with the help of the rearrangement inequality

I was trying to prove the following inequality with the help of the rearrangement inequality: $$x, y, z > 0, xyz = 1 => xy^2 + yz^2 + zx^2 \geq x+y+z.$$ The first way I came up with was to ...
Aig's user avatar
  • 5,521
2 votes
1 answer
49 views

$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$

(Crux) Let: $a,b,c>0$ such that $a+b+c=3$. Prove that: $$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$$ Solution: Put: $x=bc,y=ca,z=ab$. So the inequality can be rewritten as: $$\...
Lục Trường Phát's user avatar
4 votes
1 answer
75 views

Showing $\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\alpha_i$, for real $\alpha_i\in(0,\pi/2)$ with $\sum_{i=1}^n\cos^2\alpha_i=1$

Real numbers $\alpha_1,\ldots,\alpha_n \in \left(0,\ \frac{\pi} 2\right)$ satisfy the condition $\sum_{i=1}^n\cos^2\alpha_i=1$. Prove that $$\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\...
Mateo's user avatar
  • 4,996
10 votes
3 answers
607 views

How to prove the equations have only one real solution?

There are $n$ equations. I need answer for the case $n=3$. $$ \frac{1}{x_1}(1-x_1)^2+\frac{1}{x_2}(1-x_2)^2+\cdots+\frac{1}{x_n}(1-x_n)^2=0, $$ and $$ \frac{1}{x_1}(1-x_1^2)^j+\frac{1}{x_2}(1-x_2^2)^...
cbi's user avatar
  • 59
1 vote
4 answers
100 views

Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

If $x,y,z\ge 0: (x^2+y^2)(y^2+z^2)(z^2+x^2)=2.$ Find the maximum $$P=xy^2+yz^2+zx^2+xyz$$ I guess equality occurs when $x=y=z$ so I tried to prove homogenizing inequality $$xy^2+yz^2+zx^2+xyz \le \...
Anonymous's user avatar
-1 votes
1 answer
65 views

Inequality from Riasat Inequalitus

Let a,b,c,d>0 and $a+b+c+d=4$, prove $$\frac{4}{abcd}⩾\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$$ I found posts about this inequality, but I can't figure out where I'm wrong in my proof ...
afraidguy 's user avatar
0 votes
1 answer
42 views

Isoperimetric inequality using rearrangements

The Wikipedia post on "Symmetric decreasing rearrangements" says that "The Pólya–Szegő inequality yields, in the limit case, with $p=1$, the isoperimetric inequality". I tried to ...
UserA's user avatar
  • 411
1 vote
2 answers
84 views

Prove that $(\Sigma a^2)^5\ge27~(\Sigma a^3b^2)^2$

$\newcommand{\cyc}{\sum\limits_{\rm cyc}}$You can use \cyc for $\cyc$ in this topic. Let $a$, $b$, $c$ be positive real numbers. Prove that \[\left(a^2+b^2+c^2\...
youthdoo's user avatar
  • 1,463
1 vote
3 answers
94 views

$a,b,c\in\mathbb R^+$, prove $\frac{a^3}{bc} +\frac{b^3}{ca} + \frac{c^3}{ab} \geq \frac{a^2 + b^2}{2c} + \frac{b^2 + c^2}{2a} + \frac{c^2 + a^2}{2b}$

$a$,$b$,$c \in \mathbb R^+$, prove that $$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq \frac{a^2 + b^2}{2c} + \frac{b^2 + c^2}{2a} + \frac{c^2 + a^2}{2b}$$ ...
Ash_Blanc's user avatar
  • 1,113
2 votes
1 answer
42 views

Simple Combinatorial Optimization problem, analytical solution.

Let $P=${$p_1, ... , p_n$} with $0<p_i<1$ for all $i$, and let $f:P \rightarrow P$ be a bijective function, what is the $\arg max_f \sum_i p_i \frac{f(p_i)}{1-f(p_i)}$? Using brute force I can ...
Prof_X's user avatar
  • 23
4 votes
4 answers
153 views

Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$

Show that \begin{equation} \forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \end{equation} My approach was as ...
Martin Westin's user avatar
0 votes
2 answers
38 views

Help with the inequality $a+{k}r\leq c+\sqrt{k}s$

Let $k\geq 0$. I want to find an upper bound for $k$ given that $$a+{k}r\leq c+\sqrt{k}s$$ with $a,c,r,s$ any real constants. My attempt was to write $$(kr-\sqrt{k}s)^2\leq (c-a)^2$$ but can't go any ...
Tsk's user avatar
  • 207
3 votes
1 answer
193 views

If for each $j\in\{1,\ldots,m\},\sum_{i=1}^m {x_i}^j = \sum_{i=1}^m {y_i}^j,$ then $(x_i)^{m}_{i=1}$ is a rearrangement of $(y_i)^{m}_{i=1}.$

Related to this question, I would like to prove a stronger statement for the case where we exclude $a_i$ from the question; an alternative way to view this is to set $a_i=1$ for all $i.$ Proposition: ...
Adam Rubinson's user avatar
0 votes
1 answer
33 views

Permutation of $x_1,\ldots,x_n$ that gives least $y_1\sigma(x_1)+\ldots$ greater than $x_1y_1+\ldots:$ only two values $x_i,x_j$ must be swapped

Let $n\geq 4;\ $ let $x_1,\ldots,x_n,\ y_1,\ldots,y_n\in \mathbb{R}:\ x_i,\ i=1,\ldots,n $ are not all equal. Proposition: To find a permutation $\sigma(x_1),\ldots \sigma(x_n)\ $ of $x_1,\ldots,x_n\ $...
Adam Rubinson's user avatar
10 votes
2 answers
231 views

Find a permutation $x_{\sigma(1)},\ldots,x_{\sigma(n)}$ of $x_1,\ldots,x_n$ that maximises $\sum_{k=1}^{n-1}\vert x_{\sigma(k)}-x_{\sigma(k+1)}\vert.$

Suppose $\ x_1,\ x_2,\ \ldots,\ x_n\ $ are real numbers with $\ x_1 < x_2 <\ldots < x_n.$ Is there an efficient way to find a permutation $\ x_{\sigma(1)},\ x_{\sigma(2)},\ \ldots,\ x_{\sigma(...
Adam Rubinson's user avatar
2 votes
1 answer
130 views

Inequality $a^4 + b^4 + c^4 + k(a^3b + b^3c + c^3a)$ $\ge (k + 1)(ab^3 + bc^3 + ca^3)$ where $k\in [0,1]$

Prove the following inequality for all real numbers $a, b, c$ and for $k \in [0, 1]$: $$a^4 + b^4 + c^4 + k(a^3b + b^3c + c^3a) \geq (k + 1)(ab^3 + bc^3 + ca^3)$$ I have tried to prove by ...
MathStackExchange's user avatar
4 votes
1 answer
219 views

distance between sorted arrays

Assume we have two arrays of real numbers: $$ X = \{x_{1}, x_{2}, \dots, x_{n} \} $$ and $$ Y = \{y_{1}, y_{2}, \dots, y_{n} \} $$ Next, assume that $d = \max(|x_{i} - y_{i}|)$. Next let us sort both ...
AnTlr's user avatar
  • 99
0 votes
1 answer
196 views

Prove: $\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2\ge\sqrt{3}\sum_{cyc}{\sqrt[4]{\frac{5ab}{c}+4a}}$

Prove that the following inequality :$$\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2\ge\sqrt{3}\left(\sqrt[4]{\frac{5ab}{c}+4a}+\sqrt[4]{\frac{5bc}{a}+4b}+\sqrt[4]{\frac{5ca}{b}+4c}\right)$$ holds for all ...
Sickness's user avatar
5 votes
1 answer
227 views

Let $x\geq y\geq z>0$. Prove that $ \frac {x^{2}y}{z} + \frac {y^{2}z}{x} + \frac {z^{2}x}{y}\geq x^{2} + y^{2} + z^{2}.$

Let $x\geq y\geq z>0$. Prove that $$ \frac {x^{2}y}{z} + \frac {y^{2}z}{x} + \frac {z^{2}x}{y}\geq x^{2} + y^{2} + z^{2}.$$ The problem is from Vietnamese MO 1991 and has been posted here before. ...
Oshawott's user avatar
  • 3,956
0 votes
1 answer
43 views

Let $h_{1},h_{2},h_{3}$ be the altitudes and $m_{1},m_{2},m_{3}$ be the medians of the triangle ABC.

Show that:$$\frac{h_1}{m_1}+\frac{h_2}{m_2}+\frac{h_3}{m_3}\leq3$$ So, I was wondering if we could prevent all the hefty geometry and solve this using Chebyshev's or the Rearrangement inequality. ...
Parth Shresth's user avatar
5 votes
3 answers
270 views

Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $

Show that for positive reals $x,y,z$ the following inequality holds and that the constant cannot be improved $$ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \...
vand's user avatar
  • 113
5 votes
4 answers
256 views

Using Rearrangement Inequality .

Let $a,b,c\in\mathbf R^+$, such that $a+b+c=3$. Prove that $$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a^2+b^2+c^2}{2}$$ $Hint$ : Use Rearrangement Inequality My Work :-$\\$ Without ...
arnav_de's user avatar
  • 709
3 votes
1 answer
232 views

Is there Inequality relating the arithmetic mean (AM) and the dot product

Is there an inequality of the kind: $$ \sum_{i=1}^n a_{i}b_{i} \geq C_{1}\left(\frac{\sum_{i=1}^{n} a_{i}}{n}\right)C_{2}\left(\frac{\sum_{i=1}^{n} b_{i}}{n}\right), $$ i.e., one relating the dot (...
sdd's user avatar
  • 451
2 votes
2 answers
118 views

Prove that $a^3b^3 + b^3 + 1 \geq a^2b^2 + ab^3 + b$ (for real, positive $a$,$b$

I was working on an Olympiad-level inequality, which I was able to boil down to the following inequality: Prove that: $a^3b^3 + b^3 + 1 \geq a^2b^2 + ab^3 + b$ I think it's more useful to write it as: ...
Aditya Gupta's user avatar
1 vote
0 answers
148 views

Rearrangement inequality with constraint

For every choice of reals $$x_1 \leq \cdots \leq x_n, \quad y_1 \leq \cdots \leq y_n$$ and every permutation $$x_{\pi(1)},...,x_{\pi(n)},$$ it is known that $$x_ny_1 + \cdots + x_1y_n \leq x_{\pi(1)}...
eyestyle101's user avatar
0 votes
1 answer
44 views

Can I solve this blood question by re-arranging? [closed]

I believe the answer to the question in the attached image is the 4th bubble. Be careful not to accidentally miss the formula under the worded explanation, at the bottom of the image. The unit litre/...
questioner's user avatar
1 vote
1 answer
89 views

Given 4 numbers $a, b, c, d> 0,$ show $16\max\limits_{\bigcirc}\left \{ a^{3}+ 3bcd \right \}\!\geq\!\left ( a+ b+ c+ d \right )^{3}$

Given four positive numbers $a, b, c, d.$ Prove that $$16\max\left \{ a^{3}+ 3bcd, b^{3}+ 3cda, c^{3}+ 3dab, d^{3}+ 3abc \right \}\geq\left ( a+ b+ c+ d \right )^{3}$$ the way I think is using the ...
user avatar
2 votes
2 answers
106 views

With $n> 3,$ prove that $a_{1}+ a_{2}+ a_{3}\geq 100$ by using Karamata's inequality

Given $n$ real numbers $a_{1}, a_{2}\cdots a_{n}$ so that $$a_{1}\geq a_{2}\geq\cdots\geq a_{n}, a_{1}+ a_{2}+ \cdots+ a_{n}= 300, a_{1}^{2}+ a_{2}^{2}+ \cdots+ a_{n}^{2}> 10000$$ With $n> 3,$ ...
user avatar
1 vote
4 answers
287 views

Inequality $a^2b(a-b) + b^2c(b-c) + c^2a(c-a) \geq 0$

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that: $$a^2b(a-b) + b^2c(b-c) + c^2a(c-a) \geq 0.$$ Now, I am supposed to solve this inequality by applying only the Rearrangement ...
Hello's user avatar
  • 2,143
1 vote
1 answer
72 views

If $x,y,z>0$, $n,m\in N$, $n\ge m$ prove:$\frac{x^n}{(y+z)^m}+\frac{y^n}{(z+x)^m}+\frac{z^n}{(x+y)^m}\ge \frac{1}{2^m}(x^{n-m}+y^{n-m}+z^{n-m})$

Given positive numbers $x,y,z$ and $n,m$ positive integers with $n\ge m$ prove that: $$\frac{x^n}{(y+z)^m}+\frac{y^n}{(z+x)^m}+\frac{z^n}{(x+y)^m}\ge \frac{1}{2^m}(x^{n-m}+y^{n-m}+z^{n-m})$$ I tried ...
Michael Blane's user avatar
5 votes
1 answer
349 views

Proving $ \sum_{cyc} \frac{1}{a^6 + b^6 + 3c^3 + 4} \leq \frac{3}{3 + 2(\sqrt{ab} + \sqrt{bc} + \sqrt{ca})}$

Problem proposed for JBMO practice in symmetrical inequalities (Chebyshev, rearrangement): For every positive real numbers $a, b, c$, for which $a + b + c = 3$ we have: $$\sum_{cyc} \frac{1}{a^6 + b^...
user avatar
1 vote
0 answers
33 views

Convergence/divergence of sum of finite products of functions of reciprocals of natural numbers

Convergence/divergence of sum of finite products of functions of reciprocals of natural numbers where the product of the functions equals the identity function. Yes that was quite the mouthful. ...
Adam Rubinson's user avatar
2 votes
4 answers
121 views

prove $\sum_\text{cyc}\frac{a+2}{b+2}\le \sum_\text{cyc}\frac{a}{b}$

Prove if $a,b,c$ are positive $$\sum_\text{cyc}\frac{a+2}{b+2}\le \sum_\text{cyc}\frac{a}{b}$$ My proof:After rearranging we have to prove $$\sum_\text{cyc} \frac{b}{b^2+2b} \le \sum_\text{cyc} \frac{...
Albus Dumbledore's user avatar
1 vote
5 answers
118 views

$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$ Proof

Does anyone know hot to prove this inequality? Having: $a, b, c \gt 0$ $$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$$ I tried with the AM-GM inequality but I couldn't get ...
Royolh's user avatar
  • 99
2 votes
1 answer
85 views

Verification to solution an inequality and proving another.

I need solution verification an inequality, that I have solved because it seems too good to be true. But first, I attempted this but couldn't complete: Let $a$, $b$ and $c$ be the sides of a triangle ...
Book Of Flames's user avatar
0 votes
3 answers
161 views

Proving a cyclic inequality

Show that $a^4 + b^4 + c^4 \geq a^3b + b^3c + c^3a$ for any postive integers $a, b, c$ I'm not sure how to approach this problem. I've tried assuming that WLOG $a > b > c$ so that it is clear ...
Borntolol's user avatar
6 votes
2 answers
112 views

Why this inequality is correct

Let $0<x_1\leq\dots\leq x_m<1$, I denote $$a=\sum_{i=1}^mx_i,\qquad b=\sum_{i=1}^m\frac{1}{x_i},\qquad c=\sum_{i=1}^m\frac{x_i}{1-x_i}.$$ Im trying to prove that $$(b-m)(c+1-\frac{c}{a})\geq m(m-...
nizar's user avatar
  • 159
5 votes
4 answers
2k views

Homogenization, what it is in inequalities and how to utilize it to its fullest.

I have just learnt about homogenization, however I don't really understand it and I can't really utilize it (https://artofproblemsolving.com/wiki/index.php/Homogenization this is where I learnt about ...
user avatar
1 vote
1 answer
43 views

Rearrange 2nd sequence so that no two elements at same index are equal.

Given 2 sequences a and b what is best way to rearrange the b array so that no two elements ...
White Knife's user avatar
6 votes
3 answers
1k views

How and when to assume WLOG correctly?

Show that $$x^{2013}y+xy^{2013} \leqslant x^{2014}+y^{2014}$$ I know that this seems to be just an application of the rearrangement inequality, what I wanted to ask is that what does it actually mean ...
user avatar
0 votes
1 answer
124 views

One question and a challenge regarding Collatz looping proof.

I had a few close votes HERE on my post at mathoverflow and before I could ask this commenter to make it an answer so I could accept it, it was closed. I had ...
Fred Daniel Kline's user avatar
0 votes
2 answers
68 views

Doubt:regarding proof $\sum_{cyc}a^{2/3}b\le 3$

i had a doubt regarding the proof given in my textbook prove $$\sum_{cyc}a^{2/3}b\le 3$$ if $a,b,c>0$ and $a+b+c=3$ , the proof given is as follows $$3\sum_{cyc}a\ge \sum_{cyc}a+2\sum_{cyc}ab\ge \...
Albus Dumbledore's user avatar
4 votes
4 answers
222 views

prove $\sum_{i=1}^{n}\sqrt{a_i}\ge (n-1)\sum_{i=0}^{n}\frac{1}{\sqrt{a_i}}$ [duplicate]

prove $$\sum_{i=1}^{n}\sqrt{a_i}\ge ({n-1})\sum_{i=1}^{n}\frac{1}{\sqrt{a_i}}$$ if $$\sum_{i=1}^{n}\frac{1}{1+a_i}=1$$ My try: i tried substituiting $y_i=\frac{1}{1+a_i}$ thus $\sum y_i=1$ also ...
Albus Dumbledore's user avatar
2 votes
5 answers
95 views

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

prove $\sum_\text{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$ I couldn't proceed much. I tried rearranging the inequality and it became $a^4c+b^4a+c^4b\ge a^2b^2c+b^2c^2a+c^2a^2b.$ I tried using ...
Albus Dumbledore's user avatar
-2 votes
2 answers
150 views

Rearrangement Inequality- $\sum_{c y c} \frac{a^{2}+b c}{b+c} \geq a+b+c$

$$\sum_{c y c} \frac{a^{2}+b c}{b+c} \geq a+b+c$$ I'm confused about how to solve this. can someone give me a few hints? I'm stuck thing what even $\sum_{c y c}$ means!
Karan Lokchandani's user avatar
0 votes
1 answer
71 views

Is this Factorization?

I'm doubtful about the some parts of the solution to this question: Suppose that the real numbers $a, b, c > 1$ satisfy the condition $$ {1\over a^2-1}+{1\over b^2-1}+{1\over c^2-1}=1 $$ Prove ...
Book Of Flames's user avatar
6 votes
4 answers
2k views

If $a^2 + b^2 + c^2 = 1$, what is the the minimum value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$?

Suppose that $a^2 + b^2 + c^2 = 1$ for real positive numbers $a$, $b$, $c$. Find the minimum possible value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$. So far I've got a minimum of $\sqrt {3}$...
Boris Poris's user avatar