Log-Sobolev Inequality

My 2009 Williams College NSF “SMALL” undergraduate research Geometry Group has the following inequality for any $C^1$ function on the unit interval and for any p ≥ 1:

$\left(\int_0^1{f^{\frac{p+1}{p}}}\right)^\frac{p}{p+1}\le\int_0^1{\left(f^2+f'^2/\pi^2\right)^{1/2}}$

with equality for constant functions and if p>1 only for constant functions. They conjecture that these results still hold if $\pi^2$ on the right-hand side is replaced by $p\pi^2$ (sharp).

The case p=1 is standard and follows from Wirtinger’s Inequality.

Are any inequalities like this known?

1. LJS:

Isn’t the LHS decreasing in p, so p=1 is worst case?
This does not of course help with the conjecture.

Right you are, thanks very much. The undergraduates came upon it in a completely different way, as a generalization of the isoperimetric inequality (p=1) to sectors of planes with density $r^{p-1}$, using the four vertex theorem. Your comment inspired a simpler geometric comparison argument. Do you believe the conjecture?—FM

2. Frank Morgan:

Comment from David Thompson, SMALL undergraduate research Geometry Group ’09

The conjectured inequality says that for all $1\leq q \leq 2$,

$\left[\int_{0}^{1}f^{q}\right]^{1/q}\leq\int_{0}^{1}\sqrt{f^2+(q-1)f'^2/\pi^2}$.

The $q=1$ case holds trivially; $q=2$ is the standard isoperimetric inequality. Both sides are concave as functions of $q$. Would some kind of interpolation argument prove the result for all $q$?

3. Frank Morgan:

Francesco Maggi at Firenze notes that differentiating with respect to q at q=1 implies the following log-Sobolev inequality for uniform measure on the interval for a nonnegative function f of mean 1:

Corollary of Conjecture: $2\pi^2\int_{0}^{1}f log f\leq\int_{0}^{1}f'^2/f$.

Similarly the corollary, on small perturbations of f = 1, implies the sharp, known Wirtinger inequality:
$[\int_0^1 u^2\le(1/\pi^2)\int_0^1 (u')^2]$
for (nonperiodic) u of mean 0. (For u periodic the sharp constant is $1/4\pi^2$.)

4. Frank Morgan:

My 2012 undergraduate research Geometry Group has observed that the sharp log-Sobolev inequality on the interval follows immediately from a similar inequality on the circle.

5. Frank Morgan » Blog Archive » Manifolds with Density: Fuller References:

[…] [2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2. […]