## Log-Sobolev Inequality

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

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

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

Are any inequalities like this known?