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Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

34
votes
10answers
3k views

If both $a,b>0$, then $a^ab^b \ge a^bb^a$ [closed]

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
14
votes
2answers
369 views

Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$

Let $x_1,x_2,\ldots,x_n > 0$ such that $\dfrac{1}{1+x_1}+\cdots+\dfrac{1}{1+x_n}=1$. Prove the following inequality. $$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\dfrac{1}{\sqrt{...
12
votes
3answers
543 views

Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

The inequality: $$\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$$ Conditions: $a,b,c,d \in \mathbb{R^+}$...
10
votes
2answers
415 views

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$. Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$ For now I have only tried to write the inequality as $$4\left(\frac{...
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 ...
9
votes
1answer
211 views

Proving with AM-GM Inequality

$$\frac{4}{abcd}\geq\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$$ Given: $a+b+c+d=4$ and $a$, $b$, $c$ abd $d$ are positives. How to prove the above inequality using Arithmetic Geometric Mean ...
6
votes
5answers
316 views

Two inequalities involving the rearrangement inequality

Well, there are two more inequalities I'm struggling to prove using the Rearrangement Inequality (for $a,b,c>0$): $$ a^4b^2+a^4c^2+b^4a^2+b^4c^2+c^4a^2+c^4b^2\ge 6a^2b^2c^2 $$ and $$a^2b+ab^2+b^...
6
votes
2answers
619 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 :)
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
2answers
160 views

Let $a^4+b^4+c^4=3$. Prove that $a^2+b^2+c^2\geq a^2b+b^2c+c^2a$

Let $a^4+b^4+c^4=3$. Prove that $a^2+b^2+c^2\geq a^2b+b^2c+c^2a$ My proof :D We have the inequality $\sum_{cyc}^{ }a^2.\sum_{cyc}^{ }a\geq \sum_{cyc}^{ }a^2b $ which is equivalent to $\sum_{cyc}^{ }...
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(...
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}}{...
4
votes
4answers
138 views

Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $

Prove that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $. Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By ...
4
votes
2answers
104 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 ...
4
votes
2answers
104 views

Given $ x_1 + x_2 + … + x_{1994} = 1994$ find all $x_i$

$ x_1 + x_2 + ..... + x_{1994} = 1994$ $ x_1^3 + x_2^3 + .... +x_{1994}^3 = x_1^4 + x_2^4 + .... +x_{1994}^4$ Find all $x_i$ where all are real numbers I tried to prove all are equal to 1 using ...
4
votes
2answers
127 views

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Let $x,y,z$ be real numbers all greater than $1$, then prove that $$\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$$ My Attempt: I am trying to ...
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 ...
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 ...
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 {...
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 ...
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 ...
4
votes
0answers
2k views

If $a, b, c$ are sides of a triangle, prove that $\frac{a}{b+c-a} + \frac{b}{a+c-b} + \frac{c}{a+b-c} \ge 3$ [duplicate]

If $a, b, c$ are sides of a triangle, how can I prove that $$\frac{a}{b+c-a} + \frac{b}{a+c-b} + \frac{c}{a+b-c} \ge 3$$
3
votes
4answers
1k views

Prove the inequality $\frac{a^8+b^8+c^8}{a^3b^3c^3}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ for $a,b,c>0$

As in the title. Prove the inequality $$\frac{a^8+b^8+c^8}{a^3b^3c^3}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ for $a,b,c>0$. Thsi inequality can be proved in a pretty straightforward manner ...
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 ...
3
votes
1answer
226 views

Inequality : $ (a_1a_2+a_2a_3+\ldots+a_na_1)\left(\frac{a_1}{a^2_2+a_2}+\frac{a_2}{a^2_3+a_3}+ \ldots+\frac{a_n}{a^2_1+a_1}\right)\geq \frac{n}{n+1} $

Let $a_1, a_2, \ldots, a_n $ be positive real numbers such that $\displaystyle\sum^n_{i=1} a_i=1$. Prove that $$ (a_1a_2+a_2a_3+\ldots+a_na_1)\left(\frac{a_1}{a^2_2+a_2}+\frac{a_2}{a^2_3+a_3}+ \...
3
votes
4answers
585 views

Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$. How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$ In other words, if $a,b,c$ are non-negative real numbers, then how to prove ...
3
votes
2answers
111 views

For $a, b, c$ is the length of three sides of a triangle. Prove that $\left|\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}\right|<\frac{1}{8}$ [duplicate]

For $a, b, c$ is the length of three sides of a triangle. Prove that $$\left|\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}\right|<\frac{1}{8}$$
3
votes
2answers
560 views

How to show AM-GM inequality using rearrangement inequality?

Wikipedia article on Rearrangement inequality (link to the current revision) says (without giving any citation for this claim): Many famous inequalities can be proved by the rearrangement ...
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\...
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 ...
2
votes
3answers
165 views

Prove that $\sum\limits_{cyc} a^7 \geq \sum\limits_{cyc}a^4b^3$ [duplicate]

Prove that $a^7+b^7+c^7\ge a^4b^3+b^4c^3+c^4a^3$ SOURCE : "A Brief Introduction to Olympiad Inequalities" by Evan Chen It was one of the practice problems. Equality case is easy. I tried AM-GM ...
2
votes
5answers
112 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 ($\sin^3x$, $\...
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 ...
2
votes
3answers
146 views

How to prove that $\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$

I am trying to prove that $$\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$$ I tried some manipulations, like multiplications in $\sqrt x$ or $\sqrt y$ or using $x=\sqrt x\sqrt x$, but I'm still ...
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
4answers
55 views

let $a,b,c \in \mathbb{R^+} \ \ a+b+c =1$ Then prove that : $a^3+b^3+c^3 \geq \dfrac{1}{3}(a^2+b^2+c^2)$

Let $\{a,b,c\}\subset\mathbb{R^+}$ such that $a+b+c =1$. Prove that : $$a^3+b^3+c^3 \geq \dfrac{1}{3}(a^2+b^2+c^2)$$ $$a^3+b^3+c^3-3abc=(x+b+c)(a^2+b^2+c^2-(ab +ac+bc))$$ $$a^2+b^2+c^2=(a+b+c)^2-2(...
2
votes
4answers
180 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 ...
2
votes
3answers
122 views

Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

If $x+y+z=1$ and $x,y,z$ are positive numbers, Find the minimum of expression: $$\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$$ My solution: $$\left[\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+...
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 ...
2
votes
2answers
101 views

Inequality $\frac{3}{\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}}\geq1+ \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$ for positive $a$, $b$, $c$

If $A=\frac{3}{\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}}$ and $B = \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$ and $a,b,c>0.$ Then prove that $A\geq 1+B$ $\bf{My\; Try::}$We can write $A$ and $...
2
votes
2answers
99 views

Chebyshev Inequality toughnut [closed]

Let $a^2 + b^2 + c^2 + d^2 = 1$, where $(a,b,c,d \geq 0)$. Prove that: $$ \frac{a^{2}}{b+c+d}+\frac{b^{2}}{a+c+d}+\frac{c^{2}}{b+a+d}+\frac{d^2}{b+c+a} \geq \frac{2}{3}$$
2
votes
2answers
175 views

How prove this inequality $\sum\limits_{cyc}\frac{a^3}{b+c+d}\ge \dfrac{1}{3}$,if $\sum\limits_{cyc}a\sqrt{bc}\ge 1$

Let $a$, $b$, $c$ and $d$ be non-negative numbers such that $$a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab}\ge 1.$$ Show that $$\dfrac{a^3}{b+c+d}+\dfrac{b^3}{a+c+d}+\dfrac{c^3}{a+b+d}+\dfrac{d^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}+...
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} \...
2
votes
0answers
101 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 $n\le 16$?

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 $n \le 16$? It is ...
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 ...
1
vote
3answers
72 views

Prove that: $\sum_{i=1}^n a_i^2\geq \sum_{i=1}^n a_ib_i$

Suppose that $a_1,a_2,...,a_n$ are positive real numbers.Also assume that $b_1,b_2,...b_n$ are an arbitrary permutation of $a_i$'s. Prove that: $$\sum_{i=1}^n a_i^2\geq \sum_{i=1}^n a_ib_i.$$ We ...
1
vote
2answers
129 views

If a, b, c>0 show that $\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}$

For positive real numbers $a$, $b$, and $c$ prove that: $$\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}.$$ I let $x=\frac{a}{b}$, $y=\frac{b}{c}$, and $z=\frac{c}{a}$. Then ...
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
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 ...