  By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2
dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall
u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ for some
constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring
type problems and has interesting scaling properties. The best constant is
achieved by sign changing minimizers of a problem on periodic functions, but
does not depend on the period. Moreover, we completely characterize the
minimizers of the periodic problem.
