A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, I think the simplest way to address the mentioned request would be just to refer to the following well-known, folklor-ish, and easily proved fact:
Let $(X,\Sigma,\mu)$ be a measure space. If the function $g\colon \mathbb R\times X\to\mathbb R$ is measurable and the function $t\mapsto g(t,x)$ is log-convex for each $x\in X$, then the "mixture" function $t\mapsto\int_X g(t,x)\mu(dx)$ is log-convex as well.
However, I cannot find a reference to this fact, even when $X=\mathbb R$ and $\mu$ is the Lebesgue measure. Can you help me with this?
I only need references, not proofs. I already have references to the fact that the sum of log-convex functions is log-convex, from which the highlighted result easily follows, and actually a proof of the highlighted result is quite similar to one of the log-convexity of the sum of log-convex functions -- say, by using Hölder's inequality. However, I'd like to have a reference to the highlighted result just as stated, at least for the mentioned case when $X=\mathbb R$ and $\mu$ is the Lebesgue measure (of course, the referenced paper/book should contain a proof).
(I have posted this question on Mathematics SE, but received no answers or comments.)