Before Muirhead's Inequality can be defined, it is first necessary to define another term.
Given two sequences of real numbers and , is said to majorize if and only if all of the following are true:
With this out of the way, it is now possible to define Muirhead's Inequality.
Let be a set of positive integers, and let , and be sets of positive real numbers (note that they all contain the same number of terms) such that majorizes . Then, using sigma notation, it is possible to say that
A concrete example will probably help to understand the inequality. Since the sequence majorizes , Muirhead's inequality tells us that, for any positive , we have
A common bruteforce technique with inequalities is to clear denominators, multiply everything out, and apply Muirhead's or Schur's. However, it is worth noting that any inequality that can be proved directly with Muirhead can also be proved using the Arithmetic Mean-Geometric Mean inequality. In fact, IMO gold medalist Thomas Mildorf says it is unwise to use Muirhead in an Olympiad solution; you should use an application of AM-GM instead.
As an example, here's how the above inequality can be proved using AM-GM:
Adding these, we get
Multiplying both sides of this by (we can do this because both and are positive), we get
Muirhead's inequality is just one of a good many well-known inequalities that problem-solving students should learn.