Integrals of logarithmically concave functions

In this note we consider integrals of the form
$$ \int_Af(x,y)\,dy\stackrel{def}=I(x,A), $$
where $f$ is a finite logarithmically concave function in $E^{n+m}$ and $A$ is a convex subset of the space $E^m$. For any pair of convex sets $A$ and $B$ and any $x_1,x_2\in E^n$ we establish the inequality
$$ I(\lambda x_1+(1-\lambda)x_2,\lambda A+(1-\lambda)B)\ge I^\lambda(x_1,A)I^{1-\lambda}(x_2,B)\quad0<\lambda<1. $$