Hardy's inequality is an inequality in mathematics, named after G. H. Hardy.
Its discrete version states that if
is a sequence of non-negative real numbers, then for every real number p > 1 one has If the right-hand side is finite, equality holds if and only if
for all n. An integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then If the right-hand side is finite, equality holds if and only if f(x) = 0 almost everywhere.
Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy.
[1] The original formulation was in an integral form slightly different from the above.
The general weighted one dimensional version reads as follows:[2]: §332 if
, The general weighted one dimensional version reads as follows:[2]: §330 In the multidimensional case, Hardy's inequality can be extended to
-spaces, taking the form [3] where
is known to be sharp; by density it extends then to the Sobolev space
is an nonempty convex open set, then for every
, there exists a constant
, one has[5]: Lemma 2 Hardy’s original proof[1][2]: §327 (ii) begins with an integration by parts to get Then, by Hölder's inequality, and the conclusion follows.
A change of variables gives which is less or equal than
{\displaystyle \int _{0}^{1}\left(\int _{0}^{\infty }f(sx)^{p}\,dx\right)^{1/p}\,ds}
by Minkowski's integral inequality.
Finally, by another change of variables, the last expression equals Assuming the right-hand side to be finite, we must have
Hence, for any positive integer j, there are only finitely many terms bigger than
This allows us to construct a decreasing sequence
containing the same positive terms as the original sequence (but possibly no zero terms).
for every n, it suffices to show the inequality for the new sequence.
This follows directly from the integral form, defining
, there holds (the last inequality is equivalent to
, which is true as the new sequence is decreasing) and thus Let
be positive real numbers.
First we prove the inequality Let
-th terms in the right-hand side and left-hand side of *, that is,
We have: or According to Young's inequality we have: from which it follows that: By telescoping we have: proving *.
Applying Hölder's inequality to the right-hand side of * we have: from which we immediately obtain: Letting
we obtain Hardy's inequality.