Ricci curvature and Yamabe constants

We prove that if a closed unit volume Riemannian manifold, $(M^n,g)$, has Ricci curvature bounded from below by r>0 then the Yamabe constant of the conformal class of $g$ is at least $n\mathrm{.}r$. This inequality has already been proved by S. Ilias (Constantes explicites pour les inegalites de Sobolev sur les varietes riemannienes compactes, Ann. Inst. Fourier 33, 151-165). The equality is achieved if the metric is Einstein (with Ricci curvature r). This implies for instance that if $h$ is the Fubini-Study metric on $C{P}^2$ and $g$ is any other metric on $C{P}^2$ with $Ricci(g)\ge Ricci(h)$ then $Vol(CP^2, g) \leq Vol(CP^2, h)$.