Wirtinger's inequality for functions
For other inequalities named after Wirtinger, see Wirtinger's inequality. In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which can be viewed as certain forms of the Poincaré inequality. Theorem
There are several inequivalent versions of the Wirtinger inequality:
 Let  be a continuous and differentiable function on the interval  with average value zero and with . Then

 and equality holds if and only if  for some numbers  and . Let  be a continuous and differentiable function on the interval  with . Then

 and equality holds if and only if  for some number . Let  be a continuous and differentiable function on the interval  with average value zero. Then

 and equality holds if and only if  for some number . Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry. They can also all be regarded as special cases of various forms of the Poincaré inequality, with the optimal Poincaré constant identified explicitly. The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified. Proofs
The three versions of the Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. In each case, by a linear change of variables in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of . Fourier series
Consider the first Wirtinger inequality given above. Take  to be . Since Dirichlet's conditions are met, we can write

and the fact that the average value of  is zero means that . By Parseval's identity,

and

and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore it is seen that equality holds if and only if  for all , which is to say that . This is equivalent to the stated condition by use of the trigonometric addition formulas. Integration by parts
Consider the second Wirtinger inequality given above. Take  to be . Any differentiable function  satisfies the identity

Integration using the fundamental theorem of calculus and the boundary conditions  then shows

This proves the Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to , the general solution of which (as computed by separation of variables) is  for an arbitrary number . There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that  extends continuously to  and  for every function . This is resolved as follows. It follows from the Hölder inequality and  that

which shows that as long as

is finite, the limit of  as  converges to zero is zero. Since  for small positive values of , it follows from the squeeze theorem that  converges to zero as  converges to zero. In exactly the same way, it can be proved that  converges to zero as  converges to . Functional analysis
Consider the third Wirtinger inequality given above. Take  to be . Given a continuous function  on  of average value zero, let  denote the function  on  which is of average value zero, and with  and . From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of  are  for nonzero integers , the largest of which is then . Because  is a bounded and self-adjoint operator, it follows that

for all  of average value zero, where the equality is due to integration by parts. Finally, for any continuously differentiable function  on  of average value zero, let  be a sequence of compactly supported continuously differentiable functions on  which converge in  to . Then define

Then each  has average value zero with , which in turn implies that  has average value zero. So application of the above inequality to  is legitimate and shows that

It is possible to replace  by , and thereby prove the Wirtinger inequality, as soon as it is verified that  converges in  to . This is verified in a standard way, by writing

and applying the Hölder or Jensen inequalities. This proves the Wirtinger inequality. In the case that  is a function for which equality in the Wirtinger inequality holds, then a standard argument in the calculus of variations says that  must be a weak solution of the Euler–Lagrange equation  with , and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that  for some number . To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question. Spectral geometry
In the language of spectral geometry, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first eigenvalue and corresponding eigenfunctions of the Laplace–Beltrami operator on various one-dimensional Riemannian manifolds:
 the first eigenvalue of the Laplace–Beltrami operator on the Riemannian circle of length  is , and the corresponding eigenfunctions are the linear combinations of the two coordinate functions. the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval  is  and the corresponding eigenfunctions are given by  for arbitrary nonzero numbers . the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval  is  and the corresponding eigenfunctions are given by  for arbitrary nonzero numbers . These can also be extended to statements about higher-dimensional spaces. For example, the Riemannian circle may be viewed as the one-dimensional version of either a sphere, real projective space, or torus (of arbitrary dimension). The Wirtinger inequality, in the first version given here, can then be seen as the  case of any of the following:
 the first eigenvalue of the Laplace–Beltrami operator on the unit-radius -dimensional sphere is , and the corresponding eigenfunctions are the linear combinations of the  coordinate functions. the first eigenvalue of the Laplace–Beltrami operator on the -dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is , and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on  to the unit sphere (and then to the real projective space). the first eigenvalue of the Laplace–Beltrami operator on the -dimensional torus (given as the -fold product of the circle of length  with itself) is , and the corresponding eigenfunctions are arbitrary linear combinations of -fold products of the eigenfunctions on the circles. The second and third versions of the Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space:
 the first Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in  is the square of the smallest positive zero of the Bessel function of the first kind . the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in  is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind . Application to the isoperimetric inequality
In the first form given above, the Wirtinger inequality can be used to prove the isoperimetric inequality for curves in the plane, as found by Adolf Hurwitz in 1901. Let  be a differentiable embedding of the circle in the plane. Parametrizing the circle by  so that  has constant speed, the length  of the curve is given by

and the area  enclosed by the curve is given (due to Stokes theorem) by

Since the integrand of the integral defining  is assumed constant, there is

which can be rewritten as

The first integral is clearly nonnegative. Without changing the area or length of the curve,  can be replaced by  for some number , so as to make  have average value zero. Then the Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore

which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality , which amounts to  and then  for arbitrary numbers  and . These equations mean that the image of  is a round circle in the plane. References

 

 
 
 

Fourier analysis
Inequalities
Theorems in analysis