I am a Phd math student, and I was wondering what was the general consensus on what is publishable and what is not.

I've come here with an example in mind which should give some insight. I recently proved an inequality which was needed for some other work, and it is a very crucial inequality. The proof generalises an already well known inequality, but in a quite easy way. I mean that the generalisation is both very easy to obtain and does not use any new machinery, it's obtained by just a modification of the proof of the original one. And I mean that if the authors put a remark below their proof saying "here do this and that instead of what we did, and the generalisation is easy", everyone would see how to do it.

Now, the proof is very short and basically follows the old one, but I thought it may be worth publishing because the result, although easy to obtain, is technically new and potentially of interest (again, it's not a surprising result, but still). But the proof is not interesting, it's basically a copy. I'll talk with the supervisor to see what to do, but I was curious about different takes more experienced mathematicians have on this. Would you put the result out there (obviously saying that you used the old proof almost unchanged) or leave it be?