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, \( (\alpha)\prec (\alpha) \).

Theorem 2 (Muirhead)
 
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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