I'm familiar with Prekopa-Leindler as stated in http://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality, for example.
can one say, given $h,f,g$ that satisfy $\forall x,y , $ $h(tx+(1-t)y) \geq f(x) ^{t} g(y)^{1-t}$ that $$ \int e^{h(z)}dz \geq (\int e^{f(x)}dx)^{t} (\int e^{g(y)}dy)^{1-t} $$
