Quick description
This page contains links to articles that discuss techniques for proving inequalities.
The articles
Compressions Quick description ( Suppose that a parameter (such as volume, for instance) is associated with a certain kind of mathematical object, and one would like to find the object
for which
is maximized. A useful general idea is to find a compression: that is, an operation
that takes an object
and produces a new object
with the property that
is at least as big as
, with equality if and only if
. Then we know that any
for which
is maximized must satisfy
, which may well give us enough information about
to allow us to solve the problem. )
Proving inequalities using convexity Quick description ( Jensen's inequality states that if is a convex function and
are positive real numbers that add up to 1, then
. Many well-known inequalities follow from this one. More generally, convexity can be a powerful tool for proving inequalities. )
Double counting. Quick description ( If you have two ways of calculating the same quantity, then you may well end up with two different expressions. If these two expressions were not obviously equal in advance, then your calculations provide a proof that they are. Moreover, this proof is often elegant and conceptual. Double counting can also be used to prove inequalities, via a simple result about bipartite graphs. )
Sums of squares Quick description ( If and
are real numbers and you want to prove that
is at most as big as
, then often a good way to do it is to express
as a sum of squares of real numbers. )
The tensor power trick Quick description ( If you want to prove that , where
and
are some non-negative quantities, but you can only see how to prove a quasi-inequality that says that
for some constant
, then try to replace all objects involved in the problem by "tensor powers" of themselves and apply the quasi-inequality to those powers. If all goes well, one can show that
for all positive integers
, with a constant
which is independent of
, which implies that
since one can take
th roots and then let
tend to infinity. )
Bounding probabilities by expectations Quick description ( If is a random variable and you would like a good upper bound for the probability that
, then a good way of doing so is sometimes to choose a non-negative increasing function
and use the fact that
is at most
This article is a discussion of the method. )