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 $ϕ(t)$ defined by $$ϕ(t) = \lt(\int_V \vphi(t,x)^{-β} dx\rt)^{-\frac1{\be -n}},\qquad β\not= n,$$ where $U\subset \R$ and $V\subset \R^n$ are open bounded subsets, and $\vphi: U\times V\to \R_+$ is a $C^2-$smooth function. As a consequence, this result gives us a direct proof of the dimensional Prékopa's theorem based on a local approach.

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