Timeline for Doubly log-concave or doubly log-convex
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2022 at 15:51 | comment | added | T. Amdeberhan | Thank you, really. I drop the claim. | |
| Aug 20, 2022 at 15:47 | comment | added | Wolfgang | if you mean that in the sense that all inequalities should be strict, just replace 16 by 15.9 and/or 4 by 3.9. These are all continuous functions... | |
| Aug 20, 2022 at 15:21 | comment | added | T. Amdeberhan | @Wolfgang: you brought up an excellent point and a counter-example. Now, does it help to insist that $a_k, \mathcal{L}a_k, \mathcal{L}^2a_k$ do not have "internal zeroes"? | |
| Aug 20, 2022 at 9:08 | comment | added | Wolfgang | Take for ease of notation $a,b,c,d,e$ as consecutive terms. Suppose $b^2-ac\ge0$, $c^2-bd\ge0$, $d^2-ce\ge0$. $(a,b,c,d,e)$ is 2-fold log-concave iff $$(c^2-bd)^2\ge(b^2-ac)(d^2-ce),$$ the reciprocals are 2-fold log-convex iff $$(c^2-bd)^2\le\frac{c^2}{ae}(b^2-ac)(d^2-ce).$$ Now it seems to me that for $(1,4,8,16,31)$, the reciprocals are 2-fold log-convex but the numbers themselves not 2-fold log-concave. Have I missed something? | |
| Aug 18, 2022 at 14:41 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |