Abstract
We present a new logarithmic Sobolev inequality adapted to a log-concave measure on
\mathbb{R}
between the exponential and the Gaussian measure. More precisely, assume that
\Phi
is a symmetric convex function on
\mathbb{R}
satisfying
(1+\varepsilon)\Phi(x)\leq {x}\Phi'(x)\leq(2-\varepsilon)\Phi(x)
for
x\geq 0
large enough and with
\varepsilon\in ]0,1/2]
. We prove that the probability measure on
\mathbb{R}
\mu_\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
,
\mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi,
with
H_{\Phi}(x)= \left\{ \begin{array}{l} x^2 \text{ if }\left|x\right|
where
\Phi^*
is the Legendre–Fenchel transform of
\Phi
.