10
$\begingroup$

This problem has been posted on Math.SE but didn't receive any correct answer after a long time.


Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. Prove or disprove that $$ \left(\int_{0}^{2 \pi} \cos f(x) d x\right)^{2}+\left(\int_{0}^{2 \pi} \sqrt{(f'(x))^{2}+\sin ^{2} f(x)} d x\right)^{2} \geq(2 \pi)^{2} $$


It seems that when $f$ is an arbitrary constant, the left side equals $(2\pi)^2$ and seems to be the minimum. But how can I show that there's no other $f$ that makes the left side equal (or be less than) $(2\pi)^2$?


A geometric interpretation of the inequality has been found: Consider a closed curve on a sphere: $C=\{(\cos x\cdot\sin f(x),\,\sin x\cdot\sin f(x),\,\cos f(x))\mid x\in[0,2\pi)\}$, we have its perimeter $\displaystyle L=\int_0^{2\pi}\sqrt{(f'(x))^2+\sin^2 f(x)}\,dx$ and its area $\displaystyle S=2\pi-\int_0^{2\pi}\cos f(x)\,dx$. From spherical isoperimetric inequality $L^2\ge S\left( 4\pi-S \right)$, we have $\left( 2\pi-S \right)^2+L^2\ge\left( 2\pi \right)^2$, and the equality holds iff $C$ is any circle on the sphere. In this way we get the original inequality in the sense of geometry.

Now the question is, how to prove the inequality with only pure analysis methods?

$\endgroup$
4
  • 5
    $\begingroup$ One cannot show that 'there's no other $f$ [besides constants] that makes the left hand side equal (or be less than) $(2\pi)^2$', since it is not true. As the OP has remarked, if the corresponding closed curve $C$ is any circle on the sphere, then equality holds. Thus, there is (at least) a $3$-parameter family of functions $f$ that give equality, while the case of constant $f$ only constitutes a $1$-parameter family. $\endgroup$ – Robert Bryant Jan 23 at 13:26
  • 1
    $\begingroup$ This can be technical but nevertheless: you can always try "optimal control approach". See Section 3 here arxiv.org/pdf/1712.04590.pdf how it works. In your case due to a requirement $f(0)=f(2\pi)$ you will need to introduce one more variable $\int_{0}^{t} f'(s)ds=z$ in (3.6). Eventually the question will reduce to finding a function B of 4 variables $(t,x,y,z)$ which satisfies the corresponding Hamilton--Jacobi--Bellman PDE. See this example mathoverflow.net/questions/275980/… how it works. $\endgroup$ – Paata Ivanishvili Jan 29 at 2:08
  • $\begingroup$ Let $f(x)=\pm \arccos(1/\sqrt{1+\sin^2 x})$, with $+$ for $x<\pi$, $-$ for $x>\pi$. Then $C$ is a circle but the integral comes to $1.33252 (2\pi)^2$. What gives? See wolframalpha.com/input/… $\endgroup$ – Matt F. Jan 29 at 11:03
  • $\begingroup$ @MattF.: Actually, your $f$ does not produce an actual circle on the unit sphere, i.e., a curve that is the intersection of the sphere with a plane. That's probably why your integral isn't coming out to be $(2\pi)^2$. $\endgroup$ – Robert Bryant Jan 29 at 19:27
5
$\begingroup$

An approach that should work is to derive the differential equation that any minimizer would have to satisfy and check that its solutions are the known ones for which equality holds. To fill in the details, one would need to show that a minimizer does exist and has the necessary regularity to make the derivation of the differential equation valid, but I think that such arguments are standard in the calculus of variations and should be applicable here.

So, assume that a minimizer $f$ exists and set $$ a = \frac1{2\pi}\int_0^{2\pi}\cos f(x)\,dx\quad\text{and}\quad b = \frac1{2\pi}\int_0^{2\pi}\sqrt{f'(x)^2+\sin^2 f(x)}\ dx. $$ Note that, if $b=0$, then $f(x)$ is an integer multiple of $\pi$ and we have equality. Set this case aside and assume that $b>0$. The assumption that $f$ be a minimizer implies that if $u$ is any $2\pi$-periodic function then $I'(0)=0$ where $$ \begin{align} I(t) &= \left(\int_0^{2\pi}\cos\bigl(f(x){+}tu(x)\bigr)\,dx\right)^2\\ &\qquad\qquad+\left(\int_0^{2\pi}\sqrt{(f'(x){+}tu'(x))^2+\sin^2 (f(x){+}tu(x)\bigr)}\ dx\right)^2. \end{align} $$ Calculation (using integration by parts and the fact that $u$ and $f$ can be regarded as $2\pi$-periodic) then yields that $I'(0)/(4\pi)$ equals $$ \int_0^{2\pi} \left(-b\left(\frac{f'(x)}{\sqrt{f'(x)^2{+}\sin^2 f(x)}}\right)'+\frac{b\sin f(x)\cos f(x)}{\sqrt{f'(x)^2{+}\sin^2 f(x)}}-a\sin f(x)\right)\,u(x)\ dx. $$ Consequently, setting $\lambda = a/b$, we have that $f$ must satisfy a second-order differential equation with parameter $\lambda$ $$ \left(\frac{f'(x)}{\sqrt{f'(x)^2{+}\sin^2 f(x)}}\right)'-\frac{\sin f(x)\cos f(x)}{\sqrt{f'(x)^2{+}\sin^2 f(x)}}+\lambda\,\sin f(x) = 0.\tag1 $$ Thus, there is a $3$-parameter family of solutions to this equation with parameter, and $f$ must belong to this $3$-parameter family.

Now, it just so happens that we already know a $3$-parameter family of solutions to this equation, namely $$ f(x) = \arccos\left(\frac{u{+}(p\cos x {+} q \sin x)\sqrt{(p\cos x {+} q \sin x)^2+1-u^2}}{1{+}(p\cos x {+} q \sin x)^2}\right), \tag2 $$ where $|u|<1$ and $p$ and $q$ are $3$ real numbers, and all of these solutions $f$ give $a^2+b^2 = 1$, i.e., equality in the desired inequality. One gets other $3$-parameter families by adding an integer multiple of $\pi$ to the above formula, but these can be considered equivalent. It turns out that this then gives all of the solutions except the ones that have $\sin f(x) \equiv 0$. (Allowing $|u|\ge1$ gives families of solutions that can be defined only over subintervals of $[0,2\pi]$, and these must be taken into account in the analysis as well.)

Again, in order to make this argument fully rigorous, one has to prove that a minimizer does exist in the first place and prove a regularity result for solutions of the above ODE at places where $\sin f(x)$ vanishes. I think that those are doable, but I haven't checked the details.

$\endgroup$
2
  • 1
    $\begingroup$ Regarding the calculus of variations, these arguments indeed exist, but because the integrand of $b$ only has linear growth in $f'$, your minimizer might end up in $BV$, i.e. have jumps. $\endgroup$ – mlk Jan 29 at 12:37
  • 2
    $\begingroup$ @mlk: Indeed, there are issues that I haven't treated or discussed. Linear growth in $f'$ by itself is not really an issue: For example, the minimizers of $\int_a^b\sqrt{f'(x)^2+1}\,dx$ are all smooth, and the integrand there has linear growth. More serious is when the integrand is not strictly convex in $f'$, as it is not in this case when $\sin f(x)$ vanishes. (This is also where the equation (1) is singular, which is not a coincidence.) However, such local regularity issues arise frequently in geometrically meaningful minimization problems, and there are methods for treating them. $\endgroup$ – Robert Bryant Jan 29 at 14:43
-5
$\begingroup$

Don't trust me on this because I have never done calculus of variations (I'm still in high school), but I'm doing this just because I was bored. I also don't know how to insert special characters like pi or integral, but oh well.

Since we are trying to prove whether it is > or = 4pi² or not, we need to try our best to prove it wrong instead of right, and if we can't find a way to prove it wrong, then we are right. The best way to go about this is to find the least possible value of the left side of the equation. Let's look at the first integral. We have cos(f(x)), where f(x) is between 0 and 2pi. It can have any value between those numbers, but we want it to make the integral as small as possible. The first thing we should know is that cos(f(x)) will always have the range between -1 and 1. If we can make cos(f(x)) = 0 for [0,2pi], that will be perfect as the integral of it will be 0. In order for cos(f(x)) = 0 for [0,2pi], we need f(x) = pi/2 or 3pi/2 for [0,2pi] all the time. That means we will make f(x) always equal pi/2 or 3pi/2 (doesn't actually matter which one you use). cos(f(x)) now is cos(pi/2) or cos(3pi/2), which is just 0. Now if we take the integral from 0 to 2pi of 0, we get 0! Then we square the 0, which still gives us 0. We have made the first integral as small as possible. Now on to the next one. We have sqrt((f'(x))²+sin²(f(x))) this time. We need to remember that we have set f(x) to be a constant of pi/2 or 3pi/2, so we'll immediately plug it in. The equation is now sqrt((f'(x))²+sin²(pi/2)) which comes out to be sqrt((f'(x))²+1). Now we need to find out f'(x), which is easy because f(x) is a constant, which means f'(x) is 0. Now we have the integral from 0 to 2pi of sqrt(1) (or just 1). Now we can evaluate the integral, which comes out to be 2pi. Finally, since the integral is squared, we must square 2pi, which means it is 4pi². Now we add our two integrals, which are 4pi² and 0, and they give us 4pi² still. Hurray!! We have tried our best to get the least possible value, and our least possible value ended up with the lowest value we can have. 4pi² = 4pi².

$\endgroup$
1
  • 6
    $\begingroup$ We're glad you're interested in mathematics, and hope you keep it up. For the most part, MO is for professionals and the community expects answers at a professional level, and this answer doesn't quite reach that level, hence the down-voting and votes to delete. Don't take it too much to heart, but for future reference, please be aware of the level. $\endgroup$ – Todd Trimble Jan 29 at 16:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.