Hardy's inequality
Encyclopedia
Hardy's inequality is an inequality in mathematics
, named after G. H. Hardy
. It states that if
is a sequence
of non-negative real number
s which is not identically zero, then for every real number p > 1 one has
An integral
version of Hardy's inequality states if f is an integrable function with non-negative values, then
Equality holds if and only if
f(x) = 0 almost everywhere
.
Hardy's inequality was first published (without proof) in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, named after G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
. It states that if
is a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of non-negative real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s which is not identically zero, then for every real number p > 1 one has
An integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
version of Hardy's inequality states if f is an integrable function with non-negative values, then
Equality holds if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
f(x) = 0 almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
.
Hardy's inequality was first published (without proof) in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above.