# Questions tagged [rearrangement-inequality]

Proofs of inequalities by using Rearrangement inequality or Chebyshov inequality.

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

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 ...
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### 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. ...
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### 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 ...
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### 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. ...
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### Prove that $\sum_{cyc}\frac{m^2}{n + p^2} \ge \sum_{cyc}\frac{m}{n^2 + p}$ where $m, n, p \ge 0$ and $mnp = 1$.

Given positives $m, n, p$ such that $mnp = 1$, prove that $$\large \frac{m^2}{n + p^2} + \frac{n^2}{p + m^2} + \frac{p^2}{m + n^2} \ge \frac{m}{n^2 + p} + \frac{n}{p^2 + m} + \frac{p}{m^2 + n}$$ I ...
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### Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove $a^{3}b+ b^{3}c+ c^{3}a\leqq 8$ .

Problem. Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove: $$a^{3}b+ b^{3}c+ c^{3}a\leqq 8$$ My solution in M&Y : (and I'm looking forward to seeing ...
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### How to prove this inequality for $a,b,c>0$?

How to prove the inequality for $a,b,c>0$ : $$\frac{2a-b-c}{2(b+c)^2}+\frac{2b-a-c}{2(a+c)^2}+\frac{2c-b-a}{2(b+a)^2}\geq 0$$ ?
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### Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

$a$, $b$ and $c$ are positives such that $\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{c^2 + a^2} = 1$. Calculate the minimum value of $$\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}$$ I ...
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### Is there a problem in this statement: As $\sum_{n=1}^{\infty} \mu(n)/n = 0, \sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\infty} \mu(n)/n$.

As by Landau's proof $$\sum_{n=1}^{\infty} \mu(n)/n = 0$$ Therefore for any $N \in \mathbb{N}$, $$\sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\infty} \mu(n)/n$$ Is there a problem with the above ...
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### Given three postive numbers $a,b,c$ so that $a\geqq b\geqq c$. Prove that $\sum\limits_{cyc}\frac{a+bW}{aW+b}\geqq 3$ . [closed]

Given three postive numbers $a, b, c$ so that $a\geqq b\geqq c$. Prove that $$\sum\limits_{cyc}\frac{a+ b\sqrt{\frac{b}{c}}}{a\sqrt{\frac{b}{c}}+ b}\geqq 3$$ I make it Firstly, we need to have one ...
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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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Let $a_{i}, b_{i}, c_{i},\ d_{i}$ be non-negative sequences of length $k$ such that $$\begin{matrix} \sum_{k}a_{i} & = & nk \\ \sum_{k}b_{i} & = & nk\\ \sum_{k}c_{i} & = &... 2answers 173 views ### If a, b and c are sides of a triangle, then prove that a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c) \leqslant 3abc [duplicate] Let a, b and c be the sides of a triangle. Prove that$$a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c) \leqslant 3abc$$SOURCE: BANGLADESH MATH OLYMPIAD (Preparatory Question.) I am ... 1answer 67 views ### Inequality \frac{a_1}{1^2}+\frac{a_2}{2^2}+…+\frac{a_n}{n^2}\ge\frac{1}{1}+\frac{1}{2}+…+\frac{1}{n} [duplicate] Suppose a_i are dinstinct positive integers \forall1\le i\le n. Prove that$$\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}$$My approach: I ... 2answers 219 views ### Chebyshev's Sum Inequality Proof So I was reading up on Chebyshev's Sum Inequality, and I was a little confused about the first proof presented on Wikipedia. Specifically, the line which reads "opening the brackets". What does this ... 1answer 171 views ### Prove that \frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2} Let a,b,c\in \Bbb R^+ such that a+b+c=abc. Prove that$$\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}$$Idea 1.From a+b+c=abc\Leftrightarrow \frac{1}{ab}+... 2answers 67 views ### Algebraic inequality for positive reals a,b,c The problem is from a previous maths olympiad and the last step is to prove the inequality$$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 ...
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 ...