# Inequalities of Schur and Muirhead

Definition 1

Let $$\sum! F(a_1, \dots, a_n)$$ be the sum of $$n!$$ summands which are obtained from the function $$F(a_1, \dots, a_n)$$ making all permutations of the array $$(a_k)_{k=1}^n$$.

We will consider the special cases of the functio $$F$$, i.e. when $$F(a_1, \dots, a_n) = a_1^{\alpha_1}\cdot \cdots \cdot a_n^{\alpha_n}$$, $$\alpha_i\geq 0$$.

If $$(\alpha)$$ is an array of exponents and $$F(a_1, \dots, a_n)=a_1^{\alpha_1}\cdot \cdots \cdot a_n^{\alpha_n}$$ we will use $$T[\alpha_1, \dots,\alpha_n]$$ instead of $$\sum! F(a_1, \dots, a_n)$$, if it is clear what is the sequence $$(a)$$.

Example 1

$$T[1, 0, \dots, 0] = (n-1)!\cdot (a_1+a_2+\cdots + a_n)$$, and $$T[\frac1n, \frac1n, \dots, \frac1n]= n! \cdot \sqrt[n]{a_1 \cdot \cdots \cdot a_n}.$$

The AM-GM inequality is now expressed as:

$T[1, 0, \dots, 0] \geq T\left[\frac1n, \dots, \frac1n\right].$

Theorem 1 (Schur)

For $$\alpha\in\mathbb R$$ and $$\beta> 0$$ the following inequality holds: \begin{eqnarray*}T[\alpha+2\beta, 0, 0]+ T[\alpha, \beta, \beta] \geq 2T[\alpha+\beta, \beta, 0]. \quad\quad\quad\quad\quad (1) \end{eqnarray*}

Example 2

If we set $$\alpha=\beta=1$$, we get $x^3+y^3 + z^3 +3xyz \geq x^2y+xy^2+ y^2z+yz^2+ z^2x+zx^2.$

Definition 2

We say that the array $$(\alpha)$$ majorizes array $$(\alpha^{\prime})$$, and we write that in the following way $$(\alpha^{\prime})\prec (\alpha),$$ if we can arrange the elements of arrays $$(\alpha)$$ and $$(\alpha^{\prime})$$ in such a way that the following three conditions are satisfied:

• (i) $$\alpha_1^{\prime}+ \alpha_2^{\prime}+ \cdots + \alpha_n^{\prime} = \alpha_1 + \alpha_2 + \cdots + \alpha_n$$;

• (ii) $$\alpha_1^{\prime}\geq \alpha_2^{\prime}\geq \cdots \geq \alpha_n^{\prime}$$ i $$\alpha_1\geq \alpha_2 \geq \cdots \geq \alpha_n$$.

• (iii) $$\alpha_1^{\prime}+ \alpha_2^{\prime}+ \cdots + \alpha_{\nu}^{\prime} \leq \alpha_1 + \alpha_2 + \cdots + \alpha_{\nu}$$, for all $$1\leq \nu < n$$.

Clearly, $$(\alpha)\prec (\alpha)$$.

The necessairy and sufficient condition for comparability of $$T[\alpha]$$ and $$T[\alpha^{\prime}]$$, for all positive arrays $$(a)$$, is that one of the arrays $$(\alpha)$$ and $$(\alpha^{\prime})$$ majorizes the other. If $$(\alpha^{\prime}) \prec (\alpha)$$ then $T[\alpha^{\prime}] \leq T[\alpha].$ Equality holds if and only if $$(\alpha)$$ and $$(\alpha^{\prime})$$ are identical, or when all $$a_i$$s are equal.

Example 1 (continued)

AM-GM is now the consequence of the Muirhead’s inequality.

Problem 1

Prove that for positive numbers $$a, b$$ and $$c$$ the following equality holds: $\frac1{a^3+b^3+abc}+ \frac1{b^3+c^3+abc} + \frac1{c^3+a^3+ abc} \leq \frac1{abc}.$

Problem 2

Let $$a, b$$ and $$c$$ be positive real numbers such that $$abc=1$$. Prove that $\frac1{a^3(b+c)}+ \frac1{b^3(c+a)} + \frac1{c^3(a+b)} \geq \frac32.$

Problem 3

If $$a,b$$ and $$c$$ are positive real numbers, prove that: $\frac{a^3}{b^2-bc+c^2}+ \frac{b^3}{c^2-ca+a^2} + \frac{c^3}{a^2-ab+b^2} \geq 3 \cdot \frac{ab+bc+ca}{a+b+c}.$

Problem 4 (IMO 2005)

Let $$x,y$$ and $$z$$ be positive real numbers such that $$xyz \geq1$$. Prove that $\frac{x^5-x^2}{x^5+y^2+z^2}+ \frac{y^5-y^2}{y^5+z^2+x^2}+\frac{z^5-z^2}{z^5+x^2+y^2}\geq0.$

Problem 5

If $$a$$, $$b$$, $$c$$ are positive real numbers prove that $(a+b-c)(b+c-a)(c+a-b)\leq abc.$

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