Known proofs of Wirtinger's Inequality?

I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if $f:[0,\pi]\to \mathbb{C}$ is $C^1$ and $f(0)=f(\pi)=0$ then $$\int_0^\pi |f(t)|^2 dt \leq \int_0^\pi |f'(t)|^2 dt.$$ See link.

The proof that I know starts by proving that if $$\int_0^{2\pi} F(t) dt =0$$ then $$\int_0^{2\pi} |F(t)|^2 dt \leq \int_0^{2\pi} |F'(t)|^2 dt.$$ using Parseval's identity. From this, one proves the desired inequality for $f$ on $[0,\pi]$ by "extending" $f$ to an odd $C^1$ function on $[-\pi,\pi]$.

Are there other proofs? (Straightforward or otherwise...)

If you are willing to get a non-sharp constant, here's another proof found in many differential geometry texts. Without loss of generality assume $f \geq 0$. (Replacing $f$ by $|f|$ doesn't change the integrals on either side, if $f$ is assumed to be $C^1$.)

Let $2M = \sup f$, and let $t_0 \in (0,\pi)$ attain this maximum.

Let $X(t) = f(t) - M$ and $Y(t) = \sqrt{M^2 - X(t)^2}$ if $t \leq t_0$ and $-\sqrt{M^2 - X(t)^2}$ if $t \geq t_0$.

We have that $(X(t),Y(t))$ lies on the circle of radius $M$, and goes around the circle exactly once as $t$ goes from $0$ to $\pi$. We thus can use a well-known formula to conclude that

$$-\int_0^\pi Y(t) X'(t) \mathrm{d}t = \text{Area of disk} = \pi M^2$$

By Schwarz inequality, however, we have

$$\int_0^\pi Y(t) X'(t) \mathrm{d}t \leq \sqrt{ \int_0^\pi Y^2\mathrm{d}t \int_0^\pi X'^2\mathrm{d}t} = \sqrt{ \left(\pi M^2 - \int_0^\pi X^2\mathrm{d}t \right) \int_0^\pi X'(t)^2\mathrm{d}t }$$

Squaring we get

$$\pi^2 M^4 \leq \left(\pi M^2 - \int_0^\pi X^2 \mathrm{d}t\right) \int_0^\pi f'^2\mathrm{d}t$$

Now, notice that $$\int_0^\pi f^2 ~\mathrm{d}t = \int_0^\pi (X + M)^2 ~\mathrm{d}t = \pi M^2 + \int_0^\pi X^2 ~\mathrm{d}t + 2M \int_0^{\pi} X ~\mathrm{d}t \leq \pi M^2 (1+A)^2$$ where $$A: = \left[ \frac{1}{\pi M^2} \int_0^\pi X^2 ~\mathrm{d}t \right] < 1.$$ This implies $$\int_0^\pi f^2 ~\mathrm{d}t \leq (1 + A)^2(1-A^2) \int_0^\pi |f'|^2 ~\mathrm{d}t$$ The coefficient has a maximum when $A = 1/2$ or that $$\int_0^\pi f^2 ~\mathrm{d}t \leq \frac{27}{16} \int_0^\pi |f'|^2~\mathrm{d}t$$

If $\int_0^\pi X ~\mathrm{d}t = 0$, we can sharpen the coefficient to $(1 + A^2)(1-A^2) = 1 - A^4 \leq 1$. This can be achieved by extending $f$ to a function $g$ on $(-\pi,\pi)$ with an odd extension, exactly as you have described for the Fourier proof.

• you let $2M=$sup $f$ at first, but why you say $2f$ is bounded by $2M$ at last? – user360777 Jun 1 '17 at 8:06
• @user360777: you are right! let me see how to fix this. – Willie Wong Jun 1 '17 at 13:51

The following proof is found in section 7.7 of Hardy-Littlewood-Polya Inequalities, it is motivated by Hilbert's investigations into calculus of variations, especially Hilbert's method of invariant integrals.

Consider the expression $$(y'^2 - y^2) - (y' - y\cot x)^2 = -(1+ \cot^2 x) y^2 + 2y y' \cot x$$ So $$\left[(y'^2 - y^2) - (y' - y\cot x)^2 \right]\mathrm{d}x = -(\csc^2 x)y^2 \mathrm{d}x + 2y \mathrm{d}y \cot x = \mathrm{d} ( y^2 \cot x )$$

Now, since $y' \in L^2$, we have that $$y^2(x) = \left(\int_0^x y'(s) \mathrm{d}s\right)^2 \leq \int_0^x y'(s)^2 \mathrm{d}s \int_0^x 1\mathrm{d}s \leq x \int_0^x y'(s)^2 \mathrm{d}s$$ So we have that $$\frac{y^2(x)}{x} = o(1)$$ and hence $y = o(\sqrt{x})$. Similarly we have that $y^2$ approaches 0 superlinearly at $\pi$. This implies that $\lim_{x\to \{0,\pi\}} y^2 \cot x = 0$. Hence the exact integral

$$\int_0^\pi (y'^2 - y^2) - (y' - y\cot x)^2 \mathrm{d}x = \int \mathrm{d}\left( y^2 \cot x\right) = 0 - 0 = 0$$

Therefore we have

$$\int_0^\pi (y'^2 - y^2) \mathrm{d}x = \int_0^\pi (y' - y\cot x)^2 \mathrm{d}x \geq 0$$

with equality only if

$$y' = y \cot x$$

which is when $y = k \sin x$.

• Interesting to note that similar proofs using hypergeometric functions can be used to give analogous statements for the $L^{2n}$ versions of the inequality when $n$ is a positive integer. The general arguments are given in section 7.6 of the above mentioned book. – Willie Wong Jun 12 '12 at 15:46

You can find different proof in the book of B. Dacorogna 'Introduction to the calculus of variations'.

• Could you give for example the chapter and the number of the theorem? If the book is not available online, could you write a sketech of proof? – Davide Giraudo Dec 29 '12 at 13:44