Inequalities (Table of contents)

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 \[\sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \geq (n-1) \left( \frac1{\sqrt{x_1}}+ \frac1{\sqrt{x_2}} + \cdots + \frac1{\sqrt{x_n}}\right).\]