Difference between revisions of "Muirhead's Inequality"
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Revision as of 21:29, 20 June 2006
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
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
as desired.
Muirhead's inequality is just one of a good many well-known inequalities that problem-solving students should learn.