Timeline for Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2019 at 20:00 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added 16 characters in body
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| Aug 10, 2019 at 18:28 | vote | accept | Michael Rozenberg | ||
| Aug 10, 2019 at 18:28 | comment | added | Michael Rozenberg | It's an extremely nice proof! Thank you Peter! @GH from MO Thank you! | |
| Aug 10, 2019 at 11:49 | comment | added | GH from MO | @MichaelRozenberg: Peter Mueller proved the inequality contained in his first display. Setting $a:=n$ and $b:=1$ in this inequality, you obtain the inequality in your post. Alternatively, start from his second display, and set $y:=x^{1/2}$ and $z:=x^{(n-1)/2}$ there. | |
| Aug 10, 2019 at 11:46 | comment | added | GH from MO | @PeterMueller: Nice! | |
| Aug 10, 2019 at 11:18 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added 2 characters in body
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| Aug 10, 2019 at 11:15 | comment | added | Peter Mueller | @GHfromMO Thank you for spotting the miscalculation and fixing the proof! I added another argument for the final monotonicity assertion. | |
| Aug 10, 2019 at 11:13 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Fixed the proof
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| Aug 10, 2019 at 10:17 | comment | added | Michael Rozenberg | $\frac{x+1}{2}=\frac{x^b+1}{x^{b-1}+1}$ it's $\left(x^{b-1}-1\right)(x-1)=0,$ which for $b\neq1$ gives $x=1$ only. | |
| Aug 10, 2019 at 4:18 | comment | added | GH from MO | On the right hand side of your third display, a factor of $\frac{1}{z^2}$ is missing. That is, monotonicity is equivalent to $\left(\frac{y^{n+1}+z}{zy^{n+1}+1}\right)^{n-1}\ge \left(\frac{y^{n-1}+z}{zy^{n-1}+1}\right)^{n+1}$. I added below a detailed proof of this inequality, for any $n\in[1,\infty)$ and $y,z\in[0,1]$. | |
| Aug 9, 2019 at 23:19 | history | edited | Peter Mueller | CC BY-SA 4.0 |
fixed a typo
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| Aug 9, 2019 at 22:00 | history | edited | Peter Mueller | CC BY-SA 4.0 |
completed the proof
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| Aug 9, 2019 at 21:55 | history | edited | Peter Mueller | CC BY-SA 4.0 |
completed the proof
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| Aug 9, 2019 at 15:27 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added 119 characters in body
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| Aug 9, 2019 at 14:39 | history | edited | Peter Mueller | CC BY-SA 4.0 |
fixed a wrong statement about monotonicity
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| Aug 6, 2019 at 13:39 | history | answered | Peter Mueller | CC BY-SA 4.0 |