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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.)

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    $\begingroup$ However, I cannot find a reference to this fact I've been in this situation way too often, so I finally decided that it is not my job to look for such references and just add an Appendix with full (usually very short) proofs of any folklore stuff I need. Nobody gets hurt this way and sometimes people in the same position even cite my Appendixes later. So, in all honesty, I wouldn't bother about references because in your case the full proof fits into one line and following the reference will take the reader way more time than reading (or skipping) this line. $\endgroup$
    – fedja
    Commented Dec 13, 2017 at 21:52
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    $\begingroup$ You are quite right. This is way much easier to prove than to find a reference to, and for the reader it is much easier to read such a two-line proof than to go to the reference (even without reading the proof there). However, I'd not like to have such a proof of such a well-known fact in my paper. Also, sometimes in such cases a referee would be displeased that I prove such a well-known fact. A lose-lose situation. :-) $\endgroup$
    – Iosif Pinelis
    Commented Dec 13, 2017 at 22:04
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    $\begingroup$ I'd not like to have such a proof of such a well-known fact in my paper. Why? After all you have such a well-known name as "Iosif Pinelis" in it :lol: As to referees, if they talk nonsense, just write to the editor why you think so (I'm doing it quite often) and then it will be his headache to handle it. Don't waste your time or the time of MO folks without a really good reason! :-) $\endgroup$
    – fedja
    Commented Dec 13, 2017 at 23:09
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    $\begingroup$ Of course, I didn't want to waste other people's time. Rather, I thought someone may know such a reference right away, and then also earn some rep points for that. :-) $\endgroup$
    – Iosif Pinelis
    Commented Dec 14, 2017 at 0:12

3 Answers 3

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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.

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  • $\begingroup$ Thank you for your answer. I also found this paper, in a few days after posting this question. However, Kingman only proves that the sum of log-convex function is log convex. So, one then also needs a limit transition, however simple, to prove the result for general mixtures. I ended up with giving both a reference to Kingman and a short direct proof, using Hölder's inequality, as was suggested in the question. $\endgroup$
    – Iosif Pinelis
    Commented Jun 26, 2018 at 15:20
  • $\begingroup$ I thought in the Lemma of the first page in Kingman's paper, he proves that the space of log convex functions is closed under taking limsup. Therefore I think he has all the ingredient that is necessary. For example, with a few lines of argument you can show it works if the p.d.f. of the "mixing" distribution is either continuous or monotone. For more general things maybe there is a challenge, I haven't thought much about it. $\endgroup$
    – Ying Zhang
    Commented Jun 28, 2018 at 20:54
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    $\begingroup$ Oh, on a different note, I also find that on page 649, 16.B8, Marshall, Olkin, Arnold, "Inequalities: theory of Majorization and its applications", 2nd Edition, they show it for general mixtures and refers to Artin 1931. However, now that I look at their proof of B8, I cannot understand the last sentence when they say "A is semi pos.def. then so it its integral" -- You cannot integrate both rows at the same time without needing to prove sth more, so now I have doubt about that line. Either way, in their 16.D4, they did give an alternative proof that explicitly uses Holder's inequality. $\endgroup$
    – Ying Zhang
    Commented Jun 28, 2018 at 21:15
  • $\begingroup$ Thank you for these comments. I am now away from my desk, but am going to look at Marshall, Olkin, Arnold when I am back. $\endgroup$
    – Iosif Pinelis
    Commented Jun 29, 2018 at 0:58
  • $\begingroup$ I have taken the liberty to incorporate your latest comment into your answer, to complete it. Thank you for your help. (The proof of B8 is fine, because any mixture of positive semidefinite matrices is positive semidefinite, by the definition: $A$ is positive semidefinite if $x^TAx\ge0$ for all $x$.) $\endgroup$
    – Iosif Pinelis
    Commented Jul 4, 2018 at 14:34
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This subject (and its history) was discussed in Anosov's note in Математическое просвещение,

http://www.mathnet.ru/links/a59beea5836a0d54828088c860feecf5/mp86.pdf

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  • $\begingroup$ Thank you for your answer. However, Anosov only proves that the sum of log-convex function is log convex. So, one then also needs a limit transition, however simple, to prove the result for general mixtures. I ended up with giving both a reference to Kingman's paper mentioned elsewhere on this page and a short direct proof, using Hölder's inequality, as was suggested in the question. It indeed appears hard to find a reference for general mixtures! (??) $\endgroup$
    – Iosif Pinelis
    Commented Jun 26, 2018 at 15:24
  • $\begingroup$ The pass from the sum to a general mixture is so standard abstract nonsense (the set of log-convex functions is a convex cone closed under pointwise limits and blahblah) that I would not care at all. $\endgroup$
    – Fedor Petrov
    Commented Jun 26, 2018 at 21:44
  • $\begingroup$ I agree that this limit transition is very routine. Yet, I wanted a reference to a quite ready-to-use statement -- especially because its entire direct proof, known to me, is even simpler than the limit transition argument by itself. $\endgroup$
    – Iosif Pinelis
    Commented Jun 27, 2018 at 1:52
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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.

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    $\begingroup$ Thank you for your answer -- upvoted. The proof there only works for $X=\mathbb R$, though. $\endgroup$
    – Iosif Pinelis
    Commented Jan 31, 2023 at 15:51

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