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Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

2
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4answers
171 views

For $x+ y+ z= 3$ prove that $4\geqq x^2y+y^2z+z^2x$

Given $3$ positive real numbers $x,\,y,\,z$ and $x+ y+ z= 3$. Prove that $$4\geqq x^2y+ y^2z+ z^2x$$ This problem was homogenized so I set $x+ y+ z= 3$ to cancel stuff. Now I'm stuck. I have noticed ...
1
vote
3answers
21 views

Prove that $A_1 \cdot B_1 + A_2 \cdot B_2 > A_1 \cdot B_2 + A_2 \cdot B_1$ for $0 < A_1 < A_2$ and $0 < B_1 < B_2$

We are asked to prove or disprove that this is correct $A_1 \cdot B_1 + A_2 \cdot B_2 > A_1 \cdot B_2 + A_2 \cdot B_1$ for $0 < A_1 < A_2$ and $0 < B_1 < B_2$. I'm not very ...
3
votes
3answers
392 views

Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$ are different positive real numbers. First, I tried to simplify the proof ...
4
votes
0answers
2k views
1
vote
1answer
34 views

$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v \in \{-1,1\}^n }{sup} |\underset{i,j}{\sum} a_{ij}u_iv_j|$?

let $(a)_{ij}$ be a $M\times N$ Matrix with real entries ,is that possible to prove that: for any $x \in [-1,1]^n, y \in [-1,1]^m$ we have: $$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v ...
-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 ...
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. $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4}$$ $$\...
1
vote
1answer
73 views

How does the number $3$ help in this (probably Cauchy-based) inequality?

Given $a,b,c>0$ and $a+b+c=3$. Prove that $$\dfrac{a^2+bc}{b+ca}+\dfrac{b^2+ca}{c+ab}+\dfrac{c^2+ab}{a+bc}\ge3$$ Attempt: Using Cauchy inequality $(a+b\ge2\sqrt{ab})$: $\dfrac{a^2}{b+ca}+b+ca\...
1
vote
2answers
89 views

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3$. Prove that $$a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$$ I found a point at which the equality is attended, say $a=0,b=1,c=2$. But I ...
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{...
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+...
1
vote
3answers
40 views

Sum of cross terms vs sum of squares? [closed]

What do we know about sum of squares vs sum of cross terms? Does one always dominate the other? Any theorems on that? e.g for $a^2 + b^2 + c^2 \ < ? > \ ab + ac + bc $ for any number of ...
0
votes
1answer
52 views

Find minimum of sum of product of sequences

Let $a_{i}, b_{i}, c_{i},\ d_{i}$ be non-negative sequences of length $k$ such that $$ \begin{matrix} \sum_{k}a_{i} & = & nk \\ \sum_{k}b_{i} & = & nk\\ \sum_{k}c_{i} & = &...
0
votes
1answer
105 views

Prove $3\leqq x\sqrt{1+ y^{3}}+ y\sqrt{1+ z^{3}}+ z\sqrt{1+ x^{3}}\leqq 5$

This is an old problem of Pham Kim Hung! Prove: $$3\leqq x\sqrt{1+ y^{3}}+ y\sqrt{1+ z^{3}}+ z\sqrt{1+ x^{3}}\leqq 5$$ with $x,\,y,\,z\geqq 0$ & $x+y+z=3$ For the LHS, we have: $$\left \{ \sum\...
4
votes
2answers
73 views

If $a$, $b$ and $c$ are sides of a triangle, then prove that $a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c)$ $\leqslant$ $3abc$

Let $a$, $b$ and $c$ be the sides of a triangle. Prove that $$a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c) \leqslant 3abc$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preparatory Question.) I am ...
0
votes
2answers
109 views

Is it true that $\frac {a}{a+b}+\frac {b}{b+c}+\frac {c}{c+a}\geq \frac32$ if $a+b+c=1$?

Is it possible to prove that $ \dfrac {a}{a+b}+\dfrac {b}{b+c}+\dfrac {c}{c+a}\geq \dfrac {3}{2}$, if $ a+b+c=1, a,b,c>0$?
3
votes
1answer
66 views

Inequality $\frac{a_1}{1^2}+\frac{a_2}{2^2}+…+\frac{a_n}{n^2}\ge\frac{1}{1}+\frac{1}{2}+…+\frac{1}{n}$ [duplicate]

Suppose $a_i$ are dinstinct positive integers $\forall1\le i\le n$. Prove that $$\frac{a_1}{1^2}+\frac{a_2}{2^2}+...+\frac{a_n}{n^2}\ge\frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}$$ My approach: I ...
1
vote
2answers
65 views

Chebyshev's Sum Inequality Proof

So I was reading up on Chebyshev's Sum Inequality, and I was a little confused about the first proof presented on Wikipedia. Specifically, the line which reads "opening the brackets". What does this ...
-1
votes
5answers
50 views

Show that problem and that equality if fulfilled iff a=b

If $a, b >0$, show that $$ \frac{a}{b^2} + \frac{b}{a^2} \geq \frac{1}{a} +\frac{1}{b},$$ and that equality is fulfilled if and only if $a = b.$ I tried using elementary consequences of order ...
1
vote
1answer
148 views

Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Let $a,b,c\in \Bbb R^+$ such that $a+b+c=abc$. Prove that $$\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$$ Idea 1.From $a+b+c=abc\Leftrightarrow \frac{1}{ab}+...
0
votes
2answers
62 views

Algebraic inequality for positive reals $a,b,c$

The problem is from a previous maths olympiad and the last step is to prove the inequality $$4a^4bc + a^4c^2 + 9a^3bc^2 + 4a^3b^3 + 9a^2b^3c + a^2b^4 + 9ab^2c^3 + 4ab^4c + 4b^3c^3 + b^2c^4 +4a^3c^3 ...
4
votes
1answer
94 views

chebyshev's inequality - Question

I had a question in my exam and they asked to prove that prove that: $$3(1+a^2+a^4)\geq(1+a+a^2)^2$$ for all $a\in\mathbb R$. Now , I solved it , but the problem is that in the answer they wrote ...
1
vote
0answers
31 views

symmetrized rearrangement on sphere.

I am trying to undestand the Corollary 2.2 from Osgood, Phillips and Sarnak (see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf), that is, if $u \in W^{1}(S^...
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votes
1answer
49 views

prove the statement [closed]

Statement:If $a_1,a_2,a_3\cdots a_n$ be $n$ unequal and positive quantities and if $m>r>0$ , then $$\frac{a_1^{m}+a_2^{m}\cdots +a_n^{m}}{n}> \frac{a_1^{r}+a_2^{r}\cdots +a_n^{r}}{n}. \frac{...
2
votes
2answers
268 views

Prove the inequality using Chebyshev's Inequality

If $a,b,c \in(0,\infty)$, then prove that: $$9(a^3+b^3+c^3)\ge(a+b+c)^3$$ I was trying to prove this inequality using Chebyshev's Inequality and assuming $a\ge b \ge c$ but to no avail. Can please ...
0
votes
1answer
61 views

Prove $\sum \frac{a}{a+b^4+c^4} \le 1$

If $a,b,c \in \mathbb{R+}$ and $abc=1$ Prove That $$S=\sum \frac{a}{a+b^4+c^4} \le 1$$ My approach: we have $$S=\sum \frac{\frac{1}{bc}}{\frac{1}{bc}+b^4+c^4}=\sum \frac{1}{1+b^5c+bc^5}$$ Now by $...
9
votes
5answers
542 views

Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$

I am currently attempting to prove the following inequality $\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b} \geq \dfrac{3}{2}$ for all $ a,b,c>0$ My instinctive plan of attack is to use the AM/GM ...
4
votes
1answer
117 views

$\frac {x_1^3}{y_1}+\frac {x_2^3}{y_3}+\ldots+\frac {x_n^3}{y_n}\leq \frac {a^4+b^4}{ab (a^2+b^2)}(x_1^2+x_2^2+\ldots+x_n^2) $

Let $b>a>0$ and $x_1, x_2,\ldots,x_n,y_1, y_2,\ldots,y_n\in [a,b]$. If $$x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2\,,$$ then $$\frac {x_1^3}{y_1}+\frac {x_2^3}{y_3}+\ldots+\frac {...
1
vote
1answer
108 views

$\frac {x^3}{y}+\frac {z^3}{t}\leq \frac {a^4+b^4}{ab (a^2+b^2)}(x^2+z^2). $ [closed]

Let $0 < a < x, y, z, t < b $ s.t. $x^2+z^2= y^2+t^2$. Show that $$\frac {x^3}{y}+\frac {z^3}{t}\leq \frac {a^4+b^4}{ab (a^2+b^2)}(x^2+z^2). $$ I tried to apply Cauchy Schwartz. Also I ...
4
votes
3answers
171 views

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

If $a,b,c$ are sides of triangle Find Minimum value of $$S=\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$$ My Try: Let $$P=\sqrt{a}+\sqrt{b}+\sqrt{c}$$ we have $$S=\sum \frac{1}{\frac{\sqrt{b}}{...
6
votes
2answers
618 views

Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=4.$ Prove the inequality $$a^2bc+b^2cd+c^2da+d^2ab \leq 4 .$$ Thanks :)
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votes
0answers
36 views

Optimization of sum of squares over permutations

Suppose I have fixed, positive values $n_1, \cdots, n_\ell$ and $T$. I'm looking for an algorithm to optimize \begin{align*} f(\boldsymbol{n}) = T\left(\sum_{j=1}^{\ell}\left(\sum_{i=1}^{j}n_i\right)^...
4
votes
2answers
61 views

Application of Chebyschev inequality

I want to prove the inequality above. On the extreme ends we get a clear application of AM-GM, and I want to use the chebyschev inequality for the middle but was having trouble. My Attempt: Since ...
0
votes
1answer
142 views

Prove that inequality $\sum_{cyc} \frac{a}{b+2}\le 1$

For $a,b,c>0$ and $a^3+b^3+c^3=3$ prove that $$\frac{a}{b+2}+\frac{b}{c+2}+\frac{c}{a+2}\le 1$$ We have:$$a^3+1+1+b^3+1+1+c^3+1+1\ge 3\left(a+b+c\right)$$ $$\Rightarrow 3\left(a+b+c\right)\le a^3+...
2
votes
1answer
129 views

Prove the inequality $\left(\frac1a+\frac1b+\frac1c\right)\left(\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\right)\ge\frac9{1+abc}$

Let $a,b,c>0$. Prove the inequality $$\left(\frac1a+\frac1b+\frac1c\right)\left(\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\right)\ge\frac9{1+abc}$$ My work so far: Use AM-GM: $$\frac1{1+a}+\frac1{1+b}+...
-1
votes
3answers
127 views

Show that $a^2(a+b)^3+b^2(b+c)^3+c^2(c+a)^3\geq{\frac{8abc(a+b+c)^2}{3}}$ holds for $a,b,c>0$ [closed]

Let $a,b,c>0$. Show that $$a^2(a+b)^3+b^2(b+c)^3+c^2(c+a)^3\geq{\frac{8abc(a+b+c)^2}{3}}.$$ I have no idea. Please give a hint. Thanks!
2
votes
5answers
111 views

Rearrangement inequality and minimal value of $\dfrac{\sin^3x}{\cos x} +\dfrac{\cos^3x}{\sin x}$

For $x \in \left(0, \dfrac{\pi}{2}\right)$, is the minimum value of $\dfrac{\sin^3x}{\cos x} +\dfrac{\cos^3x}{\sin x} = 1$? So considering ($\dfrac{1}{\cos x}$, $\dfrac{1}{\sin x}$) and ($\sin^3x$, $\...
1
vote
4answers
87 views

replacing Inequalities

I encountered a problem today: Prove that: $$\frac{a^3+b^3+c^3}{a^2+b^2+c^2} \ge \frac{a+b+c}{3}$$ for all $a,b,c>0$ I used the RMS-AM inequality to replace the LHS with $$\frac{\...
-1
votes
1answer
199 views

How to prove this interesting inequality: $\frac{5x+3y+z}{5z+3y+x}+\frac{5y+3z+x}{5x+3z+y}+\frac{5z+3x+y}{5y+3x+z}\ge 3$?

Here is my question: Prove that if $x,y,z>0$ then $$\frac{5x+3y+z}{5z+3y+x}+\frac{5y+3z+x}{5x+3z+y}+\frac{5z+3x+y}{5y+3x+z}\ge 3$$ Here is my attempt: Let us assume $x>y>z$: \begin{...
0
votes
1answer
39 views

Prove that $\sum_{cyc} {\frac{y-x}{y^2-1}}$ >0

If $x,y,z$ are real numbers, each greater than 1, then show that $\frac{y-x}{y^2-1}$+$\frac{z-y}{z^2-1}$+$\frac{x-z}{x^2-1}\gt 0$ It is not the actual problem,I deducted the actual problem in those ...
4
votes
2answers
103 views

To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$

if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M G.M inequality as follows: $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get ...
2
votes
1answer
113 views

Find a maximum of: $x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $

$x,y,z \ge 0 $ , $ x+y+z =1$ Find a maximum of: $$x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $$ and when it is reached. my attempt: 1) $$x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \...
0
votes
3answers
116 views
1
vote
0answers
21 views

Generalization of the Pólya–Szegő inequality

Let $d \in \mathbb{N}, 1\leq p \leq \infty$, $f \in C^{\infty}_0 (\mathbb{R}^d)$. It is known that there is a function $g$ such that $g(x)=g_0(|x|)$ for some non-increase function $g_0:[0, \infty) ...
2
votes
2answers
119 views

Inequality : $\sum_{cyc}\frac{\sqrt{a^3c}}{2\sqrt{b^3a}+3bc}\geq \frac{3}{5}$

Let $a$, $b$ and $c$ be positive real numbers. Prove that: $$\frac{\sqrt{a^3c}}{2\sqrt{b^3a}+3bc}+\frac{\sqrt{b^3a}}{2\sqrt{c^3b}+3ca}+\frac{\sqrt{c^3b}}{2\sqrt{a^3c}+3ab}\geq \frac{3}{5}$$ My ...
4
votes
0answers
68 views

Does $(\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0$ hold for $n \leq 8$?

Let positive reals $\{a_i\}$, where not all $a_i$ are equal. Does $$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$ hold for $n \le 8$? It is ...
1
vote
1answer
99 views

How is symmetry in an inequality determined?

I was reading a book about inequalities, in that I found that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}>1\tag{1}$$ is a symmetric inequality in $a$,$b$,$c$. But if I change the order from $(a,b,c)$ to $...
5
votes
2answers
331 views

How to prove specific inequality, assuming $\prod\limits_{i=1}^{n}(a_{i}-1)=1$

Let $n\ge 2$ be postive integer and $a_{i},(i=1,2,\cdots,n)$ be real numbers such that $a_{i}>1,(i=1,2,\cdots,n),\prod_{i=1}^{n}(a_{i}-1)=1$. Show that $$\sum_{i=1}^{n}\dfrac{1}{\displaystyle\...
5
votes
1answer
100 views

Prove that $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2$

Let $x,y,z \in \mathbb{R}$ such that $x \geq y \geq z > 0$. Prove that $$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2.$$ I rearranged the inequality to get $$xy(x^2y)+yz(y^2z)+xz(...
3
votes
1answer
108 views

How to approach Inequalities that can be solved using Rearrangement Inequality?

I am learning Inequalities and today I've encountered the famous Rearrangement Inequality. I'll state the theorem: Consider any two collections of real numbers in increasing order, $$a_1\leq a_2\...