Mixtures of log-convex functions are log-convex: a reference 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.)
 A: Late to the party, those are all good points made in the comments above. I just came across a reference, Kingman1961, "A CONVEXITY PROPERTY OF POSITIVE MATRICES", The Quarterly Journal of Mathematics, Volume 12, Issue 1, 1 January 1961, Pages 283–284. 

Added by Iosif Pinelis based on a comment by Ying Zhang: For the case when $X$ is an interval (which is a completely inessential restriction), the claim of interest is proved in two different ways in Sections 16.B.8 and 16.D.4 of Marshall, Olkin, Arnold, "Inequalities: theory of Majorization and Its Applications", 2nd Edition. 
A: This subject (and its history) was discussed in Anosov's note in Математическое просвещение,
http://www.mathnet.ru/links/a59beea5836a0d54828088c860feecf5/mp86.pdf
A: For future reference, maybe one of the oldest references to such a result is the following book (page 9):

E. Artin, The gamma function. English translation of German original ``Einfuhrung in die Theorie der Gammafunktion" (Verlag B. G. Teubner, Leipzig, 1931), 1964.

