  We prove a linear inequality between the entropy and entropy dissipation
functionals for the linear Boltzmann operator (with a Maxwellian equilibrium
background). This provides a positive answer to the analogue of Cercignani's
conjecture for this linear collision operator. Our result covers the physically
relevant case of hard-spheres interactions as well as Maxwellian kernels, both
with and without a cut-off assumption. For Maxwellian kernels, the proof of the
inequality is surprisingly simple and relies on a general estimate of the
entropy of the gain operator due to Matthes and Toscani (2012) and Villani
(1998). For more general kernels, the proof relies on a comparison principle.
Finally, we also show that in the grazing collision limit our results allow to
recover known logarithmic Sobolev inequalities.
