A local proof of the dimensional Prékopa's theorem

The aim of this paper is to find an expression for second derivative of the function ϕ(t)ϕ(t) defined byϕ(t)=(∫Vφ(t,x)−βdx)−1β−n,β≠n, where U⊂RU⊂R and V⊂RnV⊂Rn are open bounded subsets, and φ:U×V→R+φ:U×V→R+ is a C2C2-smooth function. As a consequence, this result gives us a direct proof of the dimensional Prékopa's theorem based on a local approach.