Bobkov's inequality

Notation:

Let

• ${\displaystyle \gamma ^{n}(dx)=(2\pi )^{-n/2}e^{-\|x\|^{2}/2}d^{n}x}$ be the canonical Gaussian measure on ${\displaystyle \mathbb {R} ^{n}}$ with respect to the Lebesgue measure,
• ${\displaystyle \phi (x)=(2\pi )^{-1/2}e^{-x^{2}/2}}$ be the one dimensional canonical Gaussian density
• ${\displaystyle \Phi (t)=\gamma ^{1}[-\infty ,t]}$ the cumulative distribution function
• ${\displaystyle I(t):=\phi (\Phi ^{-1}(t))}$ be a function ${\displaystyle I(t):[0,1]\to [0,1]}$ that vanishes at the end points ${\displaystyle \lim \limits _{t\to 0}I(t)=\lim \limits _{t\to 1}I(t)=0.}$

There exist a generalization by Dominique Bakry und Michel Ledoux.[4]