Muirhead's Inequality states that if a sequence majorizes a sequence , then given a set of positive reals :
The inequality is easier to understand given an example. Since the sequence majorizes (as ), Muirhead's inequality states that for any positive ,
We will first prove an important fact. If we have a sequence such that , then the following result holds: for any positive reals .
Proof: By AM-GM, we know:
However, expanding the right hand side, we see or
We define our sequence such that
Therefore, we get that
We multiply this by to get:
Since , we get or
Usage on Olympiad Problems
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; one should use an application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality can be proved with AM-GM before demonstrating the full AM-GM proof.
As an example, the above inequality can be proved using AM-GM as follows:
Multiplying both sides by (as both and are positive),