Consider the function
$h(x_i, x_j) := \int_{x_i}^{\overline{z}}f(z)F(z-x_i+x_j)dz$,
where $f(z)$ is a twice continuously differentiable and strictly positive probability density function defined on $(\underline{z},\overline{z})$, $F(\cdot)$ is the corresponding cumulative distribution function, and $x_i,x_j \in (\underline{z},\overline{z})$.
I want to show (or disprove) the following:
If the probability density function $f(z)$ is log-concave on $(\underline{z},\overline{z})$, then also $h(x_i,x_j)$ is log-concave in $x_i$ for $x_i,x_j \in (\underline{z},\overline{z})$.
Two useful properties that might be exploited for this: Given that $f(z)$ is log-concave on $(\underline{z},\overline{z})$, also $F(z)$ and $1-F(z)$ are log-concave on $(\underline{z},\overline{z})$.
[As an illustration, here is the proof for $F(z)$:
Suppose that $f(z)$ is log-concave on $(\underline{z},\overline{z})$. Then by definition of log-concavity, $\frac{f'(z)}{f(z)}$ is decreasing on $(\underline{z},\overline{z})$. Hence
\begin{align*} \frac{f'(z)}{f(z)}F(z) &= \frac{f'(z)}{f(z)}\int_{\underline{z}}^{z}f(y)dy\\ &= \int_{\underline{z}}^{z}\frac{f'(z)}{f(z)}f(y)dy\\ &\leq \int_{\underline{z}}^{z}\frac{f'(y)}{f(y)}f(y)dy\\ &= \int_{\underline{z}}^{z}f'(y)dy = f(z) - f(\underline{z})\\ &\leq f(z). \end{align*}
Multiplying the inequality by $f(z) > 0$ and rearranging yields $f'(z)F(z) - f(z)^2 \leq 0$, which is equivalent to $F(z)$ being log-concave.]
I wonder whether a similar proof can be used to show my claim, however all of my attempts have remained unsuccessful.
One final remark: The exponential CDF $F(z) = 1-\exp(-z)$ has a corresponding density $f(z) = \exp(-z)$ which is exactly on the boundary of being log-concave (same for $1-F(z)$). Inserting this distribution into $h(x_i,x_j)$, a little calculation shows that in the exponential case, also $h(x_i,x_j)$ is exactly on the boundary of being log-concave. I suspect that this is meaningful for a potential proof. But I'd also be happy about any other ideas, or a potential counter-example.
Perhaps you may already have solved your problem. If not:
The function $h$ is the demand function of firm $i$ in a duopoly with independent valuations for the two goods ($x_i$ and $x_j$ are the prices). Your case satisfies the assumptions of Theorem 1 of Caplin and Nalebuff (1991) and so (by that theorem) the demand function of firm $i$ is log concave in its own price (i.e. $h$ is log-concave in $x_i$) over the region that demand is positive. (The proof of Theorem 1 relies on the the Prékopa-Borell Theorem.)