Archive for June 2009

## 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?