Wirtinger's inequality states that if 
is a real-valued absolutely
continuous function of period 
, 
is square integrable,
and
 |
(1)
|
then
 |
(2)
|
 |
(3)
|
(Hardy et al. 1988).
An endpoint form states that a real-valued absolutely continuous function 
on ![[0,1]](/images/equations/WirtingersInequality/Inline5.svg)
with 
and square-integrable derivative satisfies
 |
(4)
|
with equality iff 
. A sharp generalization to other powers is given
by the Wirtinger-Sobolev
isoperimetric constants.
Another inequality attributed to Wirtinger involves the Kähler form, which in 
can be written
 |
(5)
|
Given 
vectors 
in 
, let 
denote the oriented 
-dimensional parallelepiped
and 
its 
-dimensional volume. Then
 |
(6)
|
For 
,
equality holds iff the vectors span a 
-dimensional complex subspace
of 
with its positive complex vector space orientation.
If 
,
both sides vanish as well. Here, 
is the 
th exterior power for 
, and the orientation
of a complex subspace is determined by its complex
structure.
