# Rearrangement Inequality. Chebyshev’s Inequality

Theorem 1 (Rearrangement inequality)

If $$x_1$$, $$x_2$$, $$\dots$$, $$x_n$$ and $$y_1$$, $$y_2$$, $$\dots$$, $$y_n$$ are two non-decreasing sequences of real numbers, and if $$\sigma_1$$, $$\sigma_2$$, $$\dots$$, $$\sigma_n$$ is any permutation of $$\{1,2,\dots, n\}$$, then the following inequality holds: $x_1y_n+x_2y_{n-1}+\cdots+x_ny_1\leq x_1y_{\sigma_1}+x_2y_{\sigma_2}+\cdots+ x_ny_{\sigma_n}\leq x_1y_1+x_2y_2+\cdots+x_ny_n.$

Let us show by example how we can prove the inequality between arithmetic and geometric mean using the rearrangement inequality. We will prove it for $$n=4$$, and from there it will be clear how one can generalize the method.

Problem 1

If $$a$$, $$b$$, $$c$$, and $$d$$ are positive real numbers prove that $a^4+b^4+c^4+d^4\geq 4abcd.$

Theorem 2 (Chebyshev)

Let $$a_1\geq a_2\geq\cdots\geq a_n$$ and $$b_1\geq b_2\geq\cdots\geq b_n$$ be real numbers. Then \begin{eqnarray*} n\sum_{i=1}^n a_ib_i\geq\left(\sum_{i=1}^n a_i \right)\left(\sum_{i=1}^n b_i\right)\geq n\sum_{i=1}^n a_ib_{n+1-i}.\quad\quad\quad\quad\quad (1) \end{eqnarray*} The two inequalities become equalities at the same time when $$a_1=a_2=\cdots=a_n$$ or $$b_1=b_2=\cdots=b_n$$.

We will prove the following generalization of the above theorem. The left inequality of Theorem 2 follows by substituting $$m_i=\frac1n$$ in Theorem 3, and the right inequality follows from the application of the left inequality to the sequences $$(a_i)$$ and $$(c_i)$$ with $$c_i=-b_{n+1-i}$$.

Theorem 3 (Generalized Chebyshev’s Inequality)

Let $$a_1\geq a_2\geq\cdots\geq a_n$$ and $$b_1\geq b_2\geq\cdots\geq b_n$$ be any real numbers, and $$m_1,\dots, m_n$$ non-negative real numbers whose sum is $$1$$. Then \begin{eqnarray*} \sum_{i=1}^n a_ib_im_i\geq\left(\sum_{i=1}^n a_i m_i\right) \left(\sum_{i=1}^n b_im_i\right).\quad\quad\quad\quad\quad (2) \end{eqnarray*} The inequality become an equality if and only if $$a_1=a_2=\cdots=a_n$$ or $$b_1=b_2=\cdots=b_n$$.

Problem 1

If $$a$$, $$b$$, and $$c$$ are positive real numbers, prove the inequality $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\leq\frac{3(ab+bc+ca)}{2(a+b+c)}.$

Problem 3

Prove that the sum of distances of the orthocenter from the sides of an acute triangle is less than or equal to $$3r$$, where the $$r$$ is the inradius.

Problem 4

If $$a, b$$, and $$c$$ are the lengths of the sides of a triangle, $$s$$ its semiperimeter, and $$n\geq 1$$ an integer, prove that $\frac{a^n}{b+c}+\frac{b^n}{c+a}+ \frac{c^n}{a+b} \geq \left(\frac23\right)^{n-2} \cdot s^{n-1}.$

Problem 5

Let $$0< x_1 \leq x_2 \leq \cdots \leq x_n$$ ($$n\geq 2$$) and $\frac1{1+x_1} + \frac1{1+x_2}+ \cdots + \frac1{1+x_n} = 1.$ Prove that $\sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \geq (n-1) \left( \frac1{\sqrt{x_1}}+ \frac1{\sqrt{x_2}} + \cdots + \frac1{\sqrt{x_n}}\right).$

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