Modified logarithmic Sobolev inequalities in null curvature

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on R between the exponential and the Gaussian measure. More precisely, assume that Phi is a symmetric convex function on R satisfying (1 + epsilon)Phi(x) <= x Phi' (x) <= (2 - epsilon)Phi(x) for x >= 0 large enough and with epsilon is an element of]0, 1/2]. We prove that the probability measure on R u(Phi)(dx) = e(-Phi(x))/Z(Phi)dx satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants A, B, C > 0 such that for all smooth functions f > 0, Ent(mu Phi) (f(2)) <= A integral H-Phi ((f)/(f')) f(2)d mu(Phi), with H Phi (x) = {(x2)(Phi* (Bx)) (if vertical bar x vertical bar >= C,)/(if vertical bar x vertical bar < C), where Phi* is the Legendre-Fenchel transform of Phi.