Hardy's inequality is an
inequality in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, named after
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
.
Its discrete version states that if
is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, then for every real number ''p'' > 1 one has
:
If the right-hand side is finite, equality holds
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
for all ''n''.
An
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
version of Hardy's inequality states the following: if ''f'' is a
measurable function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
with non-negative values, then
:
If the right-hand side is finite, equality holds
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''f''(''x'') = 0
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.
Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy.
The original formulation was in an integral form slightly different from the above.
Statements
General discrete Hardy inequality
The general weighted one dimensional version reads as follows:
if
,
and
,
:
General one-dimensional integral Hardy inequality
The general weighted one dimensional version reads as follows:
* If
, then
:
* If
, then
:
Multidimensional Hardy inequalities with gradient
Multidimensional Hardy inequality around a point
In the multidimensional case, Hardy's inequality can be extended to
-spaces, taking the form
: