Timeline for Log-concavity inequality
Current License: CC BY-SA 4.0
23 events
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| Jul 30, 2020 at 14:22 | comment | added | Josiah Park | @fedja Yes, absolutely. I had an application in mind where the $z>>x$ which had me thinking otherwise. Thanks again. | |
| Jul 30, 2020 at 14:05 | comment | added | fedja | No, that's the limit: If the differences are small or you are far away from the origin, then the second derivative doesn't change noticeably and $\log\log x$ is indistinguishable from a parabola in the range you are interested in (I assume you want a uniform bound in $x,y,z,w$). | |
| Jul 30, 2020 at 13:51 | comment | added | Josiah Park | @fedja Thanks! I think it could be true for larger $c$, but I will check the details. | |
| Jul 30, 2020 at 13:49 | comment | added | fedja | The second derivative of $\log\log x$ is decreasing, so just compare it with $2$ parabolas with the same leading coefficient, in which case the optimal $c$ is just $2\sqrt{t(1-t)}$ if I haven't screwed my algebra. | |
| Jul 30, 2020 at 12:34 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 12:34 | comment | added | Josiah Park | @FedorPetrov I will adjust it to $x>1$ for this reason. | |
| Jul 30, 2020 at 12:15 | comment | added | Fedor Petrov | How do you understand the deniminators if $x<1$? | |
| Jul 30, 2020 at 11:04 | comment | added | Josiah Park | @FedorPetrov Yeah, it should, I expect. | |
| Jul 30, 2020 at 10:53 | comment | added | Fedor Petrov | May $c$ depend on $x, y, t$? | |
| Jul 30, 2020 at 10:40 | comment | added | Josiah Park | @FedorPetrov Yes, the convex combinations are within the logarithms. | |
| Jul 30, 2020 at 10:37 | comment | added | Fedor Petrov | Are the brackets missed in the numerators? | |
| Jul 30, 2020 at 9:56 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 7:17 | comment | added | Brendan McKay | Yes, it clarifies that the problem statement needs clarifying. If it is only true for some $c$, you have say "for some $c$" or "there exists $c$ such that". | |
| Jul 30, 2020 at 7:09 | comment | added | Josiah Park | @BrendanMcKay It should hold for some fixed $c$ depending on the $x,y$ and $t$. I'm interested in larger $c$. Also, if the $z$ and $w$ become spaced out enough one can find counterexamples to the inequality. Maybe this clarifies things? | |
| Jul 30, 2020 at 7:02 | comment | added | Brendan McKay | Since $c$ doesn't appear in the inequality and $|z-w|\le c|y-z|$ is true for some $c>0$, it seems that condition is irrelevant. Or did you omit an upper bound on $c$? | |
| Jul 30, 2020 at 6:42 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 6:41 | comment | added | Josiah Park | @AndrásBátkai Nope. $t$ is just fixed, while $s$ free to vary. | |
| Jul 30, 2020 at 6:40 | comment | added | András Bátkai | Is there no connection between $s$ and $t$? | |
| Jul 30, 2020 at 6:38 | history | edited | András Bátkai | CC BY-SA 4.0 |
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| Jul 30, 2020 at 6:28 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 6:22 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 6:11 | history | edited | Josiah Park | CC BY-SA 4.0 |
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| Jul 30, 2020 at 6:03 | history | asked | Josiah Park | CC BY-SA 4.0 |