# Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

92 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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)^...
2answers
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### 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 ...
1answer
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### 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 ...
1answer
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### 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 ...
3answers
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### How do I prove this inequality: $a^2+b^2+c^2 \geq ab+bc+ac$? [duplicate]

How do I prove that for any $a, b, c \in \mathbb{R}$ the inequality $a^2+b^2+c^2 \geq ab+bc+ac$ is true?
2answers
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### 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 $a+b+c=1, a,b,c>0$?
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1answer
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### 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{...
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### 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 ...
2answers
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### 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 ...
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### If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $\big(a+\frac{1}{a}\big)^2 +\big(b+\frac{1}{b}\big)^2 \ge\frac{25}{2}$ [duplicate]

If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $$\bigg(a+ \dfrac {1}{a}\bigg)^2 +\bigg(b+ \dfrac {1}{b}\bigg)^2 \ge \dfrac {25}{2}$$ My tries: I am really unable to see ...
3answers
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### 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 ...
2answers
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### How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$

Let $x>0$, $y>0$ and $z>0$. Show that $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$ I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks?
2answers
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### 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 ...
1answer
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### 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}+ \...