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I am not entirely sure if this question totally fits here. If it doesn't, I apologise in advance.

In a paper I've been working on, we have a very elegant result which, when forgetting about the abstract notions and the fancy definitions, with a lot of simplifications reduces to proving some inequalities, such as the following

For every $1\leq k \leq n-1$, the following holds: $$\frac{n(n-k+2)}{2k(\max(k,n-k)+1)} + (n-k)\left(1-\frac{1}{n-\frac{k}{2}-1}\right) \leq \frac{k(n-2)(n-k)}{(k-1)(n-1)}.$$

My issues are:

  1. The above inequality is not trivial (in some sense it is pretty tight). So saying "this can be done" and not actually doing it sounds like cheating.

  2. With a computer one can see that it holds for all small cases (say when the numbers are less than $500$), and that as you increase them the difference between the right-hand-side and the left-hand-side also increases, so it will "for sure be true" (?)

  3. Working out a proof by hand can take several hours, and a full proof might take more than a page long. Including it into a paper would possibly "ruin" the flow of ideas.

Also, I suspect that some software for "proving" such inequalities must exist. I would appreciate a general comment/advice on what to do in a situation like this one.

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    $\begingroup$ Have you considered including the proof in an appendix? That's a not too uncommon solution. $\endgroup$
    – Sam Hopkins
    Sep 18, 2021 at 19:13
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    $\begingroup$ Well, actually everything is already happening in the appendix :(. I have the main Theorem in Section 3. Its proof is contained in the Appendix A, as it involves lengthy inequalities with sums of binomial coefficients. However, these proofs, after some "clever" (but long) manipulations that allow one to get rid of the sums, reduce to three inequalities as the one I wrote in the post. $\endgroup$
    – Luis Ferroni
    Sep 18, 2021 at 19:23
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    $\begingroup$ If it's already in the appendices I don't see the issue as most people will just skim read these anyway, am I missing something? $\endgroup$
    – Hollis Williams
    Sep 18, 2021 at 19:34
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    $\begingroup$ A "page long proof" sounds like a fantastically short proof for me. If, in addition, it is in the appendix, there does not seem to be anything to worry about. $\endgroup$
    – Seva
    Sep 19, 2021 at 4:52
  • $\begingroup$ There's a chance Fedja comes along and gives a 3-line proof. $\endgroup$
    – mathworker21
    Sep 19, 2021 at 17:05

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