By variational methods, we prove the inequality $$\int_\R u''{}^2\,dx-\int_\R u''\,u^2\,dx\geq I\,\int_\R u^4\,dx\quad \forall\; u\in L^4(\R)\;\mbox{such that}\; u''\in L^2(\R) $$ 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.