TITLE: A functional inequality about log-concave functions
QUESTION [20 upvotes]: Let $f,g$ be  smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
 \int_{\mathbb{R}^{n}}  \langle \nabla f(x), x\rangle\; g  dx \cdot \int_{\mathbb{R}^{n}}  \langle \nabla g(x), x\rangle\; f  dx \leq \int_{\mathbb{R}^{n}}\langle \nabla f(x), x\rangle\ \langle \nabla g(x), x\rangle\ dx \cdot \int_{\mathbb{R}^{n}}fg\,dx\; \quad (*)  
$$
The answer seems to be positive. 
Motivation: There is an interesting question which asks whether the function $\varphi(t)=\mu(e^{t} K)$ is log-concave on $\mathbb{R}$ where $K$ is a symmetric convex body in $\mathbb{R}^{n}$ and $d\mu$ is  an even log-concave probability measure. Now if you consider the expression $(\ln \varphi(t))''_{t=0}$ and ``do integration by parts'' several times and simplify few expressions you will arrive to an inequality which follows from (*) but definitely is weaker than (*). So I was wondering if there is a simple counterexample to (*).

REPLY [2 votes]: Not sure whether it is still of interest after three years, but anyway, I suggest the following:
ensure that your assumptions on $F$ and $G$ imply
1. $x\mapsto\langle \nabla F,x\rangle$ (same for $G$) is increasing with respect to some partial order $\le$ on $\mathbb{R}^{d}$
2. $(\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}), Q, \le)$ with $Q(dx)=\frac{fg}{\langle f,g\rangle}dx$ is an FKG-space, c.f. https://projecteuclid.org/download/pdf_1/euclid.ecp/1465058067.
Hope that helps. Regards