Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

Filter by
Sorted by
Tagged with
0
votes
0answers
24 views

A variant of the rearrangement inequality for integrals?

I have troubles with verifying whether the following proposition is true or false: Let $f: [0, 1]^2 \rightarrow \mathbb{R}_+$ be a bounded function which increases with its both arguments. Let $g: [0, ...
1
vote
0answers
94 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)}...
0
votes
1answer
40 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/...
1
vote
1answer
71 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 ...
2
votes
2answers
82 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 $n> 3,$ ...
1
vote
4answers
139 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 ...
1
vote
1answer
62 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 ...
5
votes
1answer
293 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^...
0
votes
0answers
22 views

Rearrangement inequality explanation

Rearrangement inequality: For each pair of ordered real sequences $$-\infty < a_1 \leq a_2 \leq \cdots \leq a_n < \infty \text{ and } -\infty < b_1 \leq b_2 \leq \cdots \leq b_n < \infty$$ ...
1
vote
0answers
27 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. ...
2
votes
4answers
99 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{...
1
vote
5answers
86 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 ...
2
votes
1answer
72 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 ...
0
votes
3answers
47 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 ...
6
votes
2answers
104 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-...
4
votes
4answers
1k 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 ...
1
vote
1answer
17 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 ...
6
votes
3answers
817 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 ...
0
votes
1answer
82 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 ...
0
votes
2answers
52 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 \...
4
votes
4answers
162 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 ...
2
votes
5answers
74 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 ...
-2
votes
2answers
105 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!
0
votes
1answer
61 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 ...
6
votes
4answers
173 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 $\sqrt {3}$...
5
votes
1answer
68 views

on possible generalisations of $1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2$

Here is a known double inequality for positive numbers: $$ 1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2 $$ Source: https://www.cut-the-knot.org/m/Algebra/Crux4196.shtml I'm curious if at least ...
2
votes
2answers
92 views

Prove: $\sqrt{\frac{bc}{a(3b+a)}} + \sqrt{\frac{ac}{b(3c+b)}} + \sqrt{\frac{ab}{c(3a+c)}} \ge \frac{3}{2}$.

Prove: $\sqrt{\dfrac{bc}{a(3b+a)}} + \sqrt{\dfrac{ac}{b(3c+b)}} + \sqrt{\dfrac{ab}{c(3a+c)}} \ge \dfrac{3}{2}$ with $a, b, c$ are positive real numbers. Let $a \le b \le c$: \begin{align*} \sqrt{\...
2
votes
4answers
77 views

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$? I tried substitution on the bottom (for example $\frac{b-1}{\frac{1}{a}+1}$), but I then a very similar ...
1
vote
1answer
49 views

Maximum value of a parameter in a cyclic inequality?

I want to figure out the set of possible values for $\alpha$ so that the inequality \begin{align*} \sum_{i=1}^{n}x_i^2 + 2\alpha\sum_{i=1}^{n-1}x_{i}x_{i+1} \ge 0 \end{align*}
5
votes
5answers
164 views

If $a, b, c\in\mathbb R^+, $ then prove that $a^3b+b^3c+c^3a\ge abc(a+b+c) .$

While trying to prove it, I proved the following two inequalities: $a^4+b^4+c^4\ge abc(a+b+c)$ and $(a^2b+b^2c+c^2a)(ab+bc+ca)\ge abc(a+b+c)^2.$ The later one, on some simplification gives $a^3b+b^...
1
vote
1answer
49 views

If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$.

Question: If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$. Solution: Observe that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\...
1
vote
2answers
41 views

Proving an inequality regarding increasing or decreasing sequences

Suppose $a_1 \geq a_2 $ and $b_1 \geq b_2$. Prove that $$ (a_1+a_2)(b_1+b_2) \leq 2 (a_1b_1 + a_2 b_2) $$ Generalize sol attempt: We can write $a_1 b_1 + a_2 b_2 + a_2 b_1 + a_1 b_2 \leq 2a_1 b_1 + 2 ...
4
votes
2answers
108 views

If $a+b+c=3$ Prove that $a^{2}+b^{2}+c^{2}\geq\frac{2+a}{2+b}+\frac{2+b}{2+c}+\frac{2+c}{2+a}$

Question - Let $a, b, c$ be positive real numbers such that $a+b+c=3 .$ Prove that $$ a^{2}+b^{2}+c^{2} \geq \frac{2+a}{2+b}+\frac{2+b}{2+c}+\frac{2+c}{2+a} $$ My try - i tried putting $a+2 = x,...
0
votes
2answers
57 views

$ \frac{a^{2}}{a+2 b^{3}}+\frac{b^{2}}{b+2 c^{3}}+\frac{c^{2}}{c+2 a^{3}} \geq 1 $

Question Let.a, $b, c$ be positive real numbers with sum 3 . Prove that $$ \frac{a^{2}}{a+2 b^{3}}+\frac{b^{2}}{b+2 c^{3}}+\frac{c^{2}}{c+2 a^{3}} \geq 1 $$ my doubt - by using cauchy reverse ...
2
votes
1answer
44 views

Where is the flaw in this using Rearrangement Inequality?

I assume that the reader understands by Rearrangement Inequality that if $a_i$ and $b_i$ are reals, and $a_1 ≤ a_2 ≤ ... ≤ a_n $ and $b_1 ≤ b_2 ≤ ...≤ b_n$ then $\Sigma_{i=0}^{i=n} a_i × b_i$ ≥ $\...
1
vote
1answer
102 views

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$ prove that $$\frac{4}{a+3b}+\frac{4}{b+3c}+\frac{4}{c+3a}\geq\frac{3}{a+2b}+\frac{3}{b+2c}+\frac{3}{c+2a}.$$ I can't really find a way to exploit the ...
2
votes
1answer
80 views

Inequality from Israel TST

Let $a, b, c, d$ be nonnegative numbers such that $a+b+c+d=18.$ Prove that: $$\sqrt{\frac{a}{b+6}}+\sqrt{\frac{b}{c+6}}+\sqrt{\frac{c}{d+6}}+\sqrt{\frac{d}{a+6}}\leq5\sqrt{\frac{2}{7}}$$ These are my ...
1
vote
1answer
110 views

One of my old inequality (very sharp)

I'm proud to present one of my old inequality that I can't solve : Let $a,b,c>0$ such that $a+b+c=1$ and $a\ge b \geq c $ then we have :$$\sqrt{\frac{a}{a^a+b^b}}+\sqrt{\frac{b}{b^b+c^c}}+\...
0
votes
1answer
41 views

How many Steiner Symmetrizations does it take to make an arbitrary set convex?

I have not seen this question investigated before but I might be wrong: Can any subset of $\mathbb{R}^d$ be turned into a convex set by finitely many steiner symmetrizations? If yes, is the number of ...
0
votes
2answers
50 views

Natural numbers inequality $na^{n-1}b\leq(n-1)a^n+b^n$ by induction

Let $a$ and $b$ be arbitrary natural numbers and $n$ some positive integer. How to prove the inequality $$na^{n-1}b\leq(n-1)a^n+b^n$$ by induction for all $n$? This is related to this result, and, ...
1
vote
2answers
86 views

$\frac{a}{a+2b+c}+\frac{b}{b+2c+a}+\frac{c}{c+2a+b}\geq\frac{3}{4}$

I found the following exercise: Prove that $$\frac{a}{a+2b+c}+\frac{b}{b+2c+a}+\frac{c}{c+2a+b}\geq\frac{3}{4}$$ for any positive $a$, $b$, $c$. I tried substituting the denominators but it led me ...
0
votes
0answers
21 views

Can I do this? Divide the terms inside $ (\frac {(m-a)^2}{2b} + d)^N $ by $d$?

For part $d)$ of this question I began with the equation in part c and divided all terms within the bracket by $d$. I then used the substitution $d = \frac {c}{E[r]}$ and arrived at the stated answer. ...
0
votes
2answers
56 views

Proving an inequality involving fractions and square roots holds

I have tried to prove the following inequality holds using a few approaches but none have worked. I am not really sure if I'm missing something. Here's the question: For every $x, y > 0$ prove ...
0
votes
0answers
30 views

Rearrangement inequality with infinite variables

Is the rearrangement inequality generalizable for infinite number of variables? In other words if I have an infinite sum $a_1x_1 + a_2x_2 + ... $ And we can change the order of the coefficients. ...
0
votes
1answer
84 views

Largest integer $k$ such that $\frac{a^{m+1}+b^{m+1}}{a^m+b^m}\geq\sqrt[k]{\frac{a^k+b^k}2}$

The setup is as follows: Suppose that $m$ is a given natural number. What is the greatest natural number $k$ such that for all real numbers $a,b>0$, we have $$\sqrt[k]{\frac{a^k+b^k}2}\le\frac{a^{...
0
votes
3answers
80 views

The inequality $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\ge \frac{3}{2}$ [duplicate]

I was told that the following inequality $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\ge \frac{3}{2}$$ can be solved by the rearrangement inequality https://en.wikipedia.org/wiki/...
0
votes
0answers
38 views

An elegant but difficult inequality. [duplicate]

$$\frac{x}{\sqrt{x+y}} + \frac{y}{\sqrt{y+z}} + \frac{z}{\sqrt{z+x}}$$ Over all non-negative $x,y,z$ satisfying $x+y+z=4$, let the maximum value of the above expression be $M$. What is the value of $...
10
votes
2answers
236 views

Typical Olympiad Inequality? If $\sum_i^na_i=n$ with $a_i>0$, then $\sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n$

Let $\sum_i^na_i=n$, $a_i>0$. Then prove that $$ \sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n $$ I have tried AM-GM, Cauchy-Schwarz, Rearrangement etc. but nothing seems to work. The ...
6
votes
2answers
242 views

Inequality : $\frac{a}{\exp(a+b)}+\frac{b}{\exp(b+c)}+\frac{c}{\exp(c+a)}\leq \exp\Big(\frac{-2}{3}\Big)$

It's a charming problem : Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$\frac{a}{\exp(a+b)}+\frac{b}{\exp(b+c)}+\frac{c}{\exp(c+a)}\leq \exp\Big(\frac{-2}{3}\Big)$$ I know the identity : ...
5
votes
3answers
242 views

Nice olympiad inequality :$\frac{xy^2}{4y^3+3}+\frac{yz^2}{4z^3+3}+\frac{zx^2}{4x^3+3}\leq \frac{3}{7}$

I have this to solve : Let $x,y,z>0$ such that $x+y+z=3$ then we have : $$\frac{xy^2}{4y^3+3}+\frac{yz^2}{4z^3+3}+\frac{zx^2}{4x^3+3}\leq \frac{3}{7}$$ I try to use Jensen's inequality but ...