Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. 
Its discrete version states that if formula_1 is a sequence of non-negative real numbers, then for every real number "p" ; 1 one has
formula_2
If the right-hand side is finite, equality holds if and only if formula_3 for all "n".
An integral version of Hardy's inequality states the following: if "f" is a measurable function with non-negative values, then
formula_4
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. 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 formula_5, formula_6 and formula_7,
formula_8
General one-dimensional integral Hardy inequality.
The general weighted one dimensional version reads as follows:
formula_10 
formula_12
Multidimensional Hardy inequalities with gradient.
Multidimensional Hardy inequality around a point.
In the multidimensional case, Hardy's inequality can be extended to formula_13-spaces, taking the form 
formula_14
where formula_15, and where the constant formula_16 is known to be sharp; by density it extends then to the Sobolev space formula_17.
Similarly, if formula_18, then one has for every formula_15
formula_20
Multidimensional Hardy inequality near the boundary.
If formula_21 is an nonempty convex open set, then for every formula_22,
formula_23
and the constant cannot be improved.
Fractional Hardy inequality.
If formula_24 and formula_25, formula_26, there exists a constant formula_27 such that for every formula_28 satisfying formula_29, one has
formula_30
Proof of the inequality.
Integral version (integration by parts and Hölder).
Hardy’s original proof begins with an integration by parts to get
formula_31
Then, by Hölder's inequality,
formula_32
and the conclusion follows.
Integral version (scaling and Minkowski).
A change of variables gives 
formula_33
which is less or equal than formula_34 by Minkowski's integral inequality. 
Finally, by another change of variables, the last expression equals 
formula_35
Discrete version: from the continuous version.
Assuming the right-hand side to be finite, we must have formula_36 as formula_37. Hence, for any positive integer "j", there are only finitely many terms bigger than formula_38. 
This allows us to construct a decreasing sequence formula_39 containing the same positive terms as the original sequence (but possibly no zero terms). Since formula_40 for every "n", it suffices to show the inequality for the new sequence. This follows directly from the integral form, defining formula_41 if formula_42 and formula_43 otherwise. Indeed, one has 
formula_44 
and, for formula_42, there holds 
formula_46
(the last inequality is equivalent to formula_47, which is true as the new sequence is decreasing) and thus 
formula_48.
Discrete version: Direct proof.
Let formula_7 and let formula_50 be positive real numbers. Set formula_51. 
First we prove the inequality 
Let formula_52 and let formula_53 be the difference between the formula_54-th terms in the right-hand side and left-hand side of *, that is, formula_55. We have: 
formula_56
or
formula_57
According to Young's inequality we have:
formula_58
from which it follows that:
formula_59
By telescoping we have:
formula_60
proving *. 
Applying Hölder's inequality to the right-hand side of * we have:
formula_61
from which we immediately obtain:
formula_62
Letting formula_63 we obtain Hardy's inequality.