Precursor to Prekopa-Leindler

epsilon emperor

For $f: \mathbb R^n \to \mathbb R$, and fixed $z \in\mathbb R$, we define $fz: \mathbb R^{n-1}\to \mathbb R$ such that $fz(x) = f(x,z)$ for every $x \in \mathbb R^{n-1}$. I have 3 functions, $f,g,h : \mathbb R^n \to \mathbb R$, such that $$h((1-t)x + ty) \ge (f(x))^{1-t} (g(y))^{t}$$ for every $x,y\in \mathbb R^n$. I want to show that $$\int{\mathbb R^{n-1}} h{(1-t)x + ty} \ge \left( \int{\mathbb R^{n-1}} fx \right)^{1-t} \left(\int{\mathbb R^{n-1}} gy \right)^t$$ for every $x,y \in\mathbb R$. How do I do this? Here, $f,g,h \ge 0$ and measurable. Also, $t \in (0,1)$.

Friday, 14 May 2021 07:06 GMT