Timeline for A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| May 11, 2022 at 21:08 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| May 11, 2022 at 7:16 | history | edited | MathArt | CC BY-SA 4.0 |
added 78 characters in body; edited title
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| May 8, 2022 at 12:21 | answer | added | Christophe Leuridan | timeline score: 0 | |
| May 7, 2022 at 18:10 | vote | accept | MathArt | ||
| May 7, 2022 at 16:55 | answer | added | Christian Remling | timeline score: 3 | |
| May 7, 2022 at 15:52 | comment | added | Andrea Marino | If you write it as $\frac{f'}{1-f^2} - \frac{1}{1-t^2}$, you can rewrite the espression as $D_s \log \frac{ K(f(s)) }{K(s) }\ge 0$, where $K(x) =\frac{1+x}{1-x}$. This is equivalent to $D_s \frac{ K(f(s)) }{K(s) }\ge 0$. | |
| May 7, 2022 at 15:08 | comment | added | username | @AnthonyQuas not concave increasing.. | |
| May 7, 2022 at 14:43 | comment | added | Anthony Quas | Isn’t your inequality false for the piecewise linear function through $(0,0)$, $(\frac 12,\frac 34)$, $(1,0)$ and hence for anything smooth nearby? | |
| May 7, 2022 at 14:43 | comment | added | Andrea Marino | This approach shows that a minimiser (if it exists) $f$ satisfies $f(x) = \frac{-u'(x) (1-x^2) -u(x) ^2}{2u(x) }$ in all $x$ such tha $u(x)\neq 0$, for all $u$ such that $f+u \in \textrm{YourSpace}$. Suppose now that by contradiction $f$ is $\neq 0,1$ in some point (so in an interval $(a, b) $). Consider the $u(x) $ that is constantly $\epsilon$ in a subinterval $(a', b') $ and $0$ outside $(a, b) $. You obtain that $f(x) \equiv \pm \epsilon/2 $ for all $\epsilon$ small enough. Thus the only candidates $f$ are the constant func. 0,1 which does not belong to your space. Thus there is no min. | |
| May 7, 2022 at 14:18 | comment | added | Andrea Marino | I am not fresh enough to produce an effective calculation, but what about trying to minimize the functional $f'(x) (1-x^2) -(1-f(x) ^2) $ over the space of concave increasing diffeomorphisms? | |
| May 7, 2022 at 13:53 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
edited tags
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| May 7, 2022 at 12:33 | history | asked | MathArt | CC BY-SA 4.0 |