I have an explicit pdf $p(x)$ that I want to show is log-concave, i.e. $\log p(x)$ is a concave function. Under suitable assumptions, this is equivalent to showing that $p(x)$ satisfies the inequality
$$p''(x)p(x) \leq (p'(x))^2.$$
In my situation, $p(x) = \mathbb{E}[f(x,t)]$, where $t$ is uniform in some compact convex set. Differentiating under the integral sign, I can reduce to showing that
$$\mathbb{E}[\partial_x^2 f(x,t)]\mathbb{E}[f(x,t)] \leq \mathbb{E}[\partial_x f(x,t)]^2$$
i.e. I need to bound the product of integrals by another (related) product of integrals.
What tools are there for this? There are many inequalities for bounding the integral of products (for example, Holder's, and variants). I am less familiar with what options there are in this slightly different setting.
If people are particularly interested, $f(x,t) = \frac{\exp(\frac{-x^2}{\lVert \vec t\rVert_2^2})}{\lVert \vec t\rVert_2}$, and we are integrating over a (centrally symmetric) convex body $K\subseteq \mathbb{R}^n$, i.e. this is all secretly a multivariable problem (though I am trying to establish log-concavity in terms of the scalar parameter $x$).
So far, I have tried writing $(\int f(y)d\vec y)(\int g(z)d\vec z) = \int f(y)g(z)d\vec y d\vec z$ as a product of integrals over independent variables, and then tried seeing if I could get pointwise bounds on the integrands $f(y)g(z)$, which would immediately imply my desired bound. These pointwise bounds don't hold for all parameters though --- in particular, I end up requiring that $\lVert \vec z\rVert_2^2 \geq \frac{4k^2}{4k^2-2}$ is bounded away for zero (for one of the variables in the above argument --- there are no conditions on the second variable), but in my applications the integration region is convex, i.e. I am integrating over a region that includes $\lVert \vec z\rVert_2 = 0$, so I cannot assume this bound.