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. 
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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](https://math.stackexchange.com/questions/2563457/mixtures-of-log-convex-functions-are-log-convex-a-reference), but received no answers or comments.)



Mixtures of log-convex functions are log-convex: a reference
Late to the party, those are all good points made in the comments above. I just came across a reference, [Kingman1961](https://academic.oup.com/qjmath/article-abstract/12/1/283/1534893?redirectedFrom=PDF), "A CONVEXITY PROPERTY OF POSITIVE MATRICES", The Quarterly Journal of Mathematics, Volume 12, Issue 1, 1 January 1961, Pages 283–284. 




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**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. 


