What is majorization, and how is it related to Muirhead's inequalities? I was reading the following article: http://math.rice.edu/MathCircle/References/0%20Mildorf%20Inequalities%202%20-%20LECTURE.pdf
As I'm not too experienced with problem solving in inequalities, I have a few questions about this. 


*

*What is majorization? I don't really understand this - especially the application to Muirhead's inequality.

*What does $\sum_{sym}$ mean? How does Muirhead's inequality work? Could I have an example, please? 
 A: *

*The definition of marjorization is straightforward. For two vectors $\mathbf{x}=(x_1,\dots,x_n)$ and $\mathbf{y}=(y_1,\dots,y_n)$ with non-increasing terms, then $\mathbf{x}$ is said to majorize $\mathbf{y}$ if
$$
\sum_{j=1}^k x_j \geq \sum_{j=1}^k y_j  \qquad (k=1,2,\dots,n).
$$
As an example, $(3,2,1)$ is majorized by $(4,2,1)$ since
$$4>3, \qquad 4+2>3+2 \qquad\text{and}\qquad 4+2+1>3+2+1.$$
Additionally, $(3,2,1)$ is majorized by $(5,2,0)$ but not by $(3,0,0)$. If you want to compare two vectors whose elements may not be non-increasing, first you sort them. For further details see the technical report Inequalities via Majorization, which appears to be notes leading to this book. There's also a Wikipedia page.

*Let's look at an example of the operator $\sum_{\text{sym}}$ when applied to the monomial $x^2y^3z^5$. We expand using the permutations of the set of exponents $\{2,3,5\}$. This results in six terms:
$$
\sum_{\text{sym}} x^2y^3z^5 = x^2y^3z^5 + x^2y^5z^3 + x^3y^2z^5 \\ + x^5y^3z^2 + x^5y^2z^3 + x^3y^5z^2.
$$
The Wikipedia article on Muirhead's inequality has two basic applications. For instance, to prove $x^2+y^2 \geq 2xy$, write it as
$$
\color{blue}{x^2+y^2=} \sum_{\text{sym}} x^2y^0 \geq \sum_{\text{sym}} x^1y^1 \color{blue}{=xy+xy}
$$
and observe that $(2,0)$ majorizes $(1,1)$. Note here that we consider the repeated exponents on the right hand side as distinct.

