Timeline for Is the following function decreasing on $(0,1)$?
Current License: CC BY-SA 3.0
9 events
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| Jul 24, 2012 at 14:48 | comment | added | Malik Younsi | @juan Yes, I tried that already, without much success unfortunately. Anyway, thank you for the suggestion. | |
| Jul 24, 2012 at 10:24 | comment | added | juan | There exists a power series $q = a_2 k^2+\cdots$, after changing $k$ into $4k$ $a_n$ in this expansion is A005797 in the OEIS. Then your $f(k)$ is $(1-q)\vartheta_2^2/\sqrt{q}$. After substituting $q$ by this series. This may be a way to get your property. I do not say this is easy. Certainly in this way dissapear your terrible looking hyperbolic sin. | |
| Jul 23, 2012 at 21:06 | comment | added | Malik Younsi | @GH: Yes indeed, I'm talking about the function $f(k)$ in the question. | |
| Jul 23, 2012 at 20:56 | comment | added | GH from MO | @Malik: That is an interesting observation, and I have no idea how to explain it. Just to clarify: you talk about your $f(k)$, not juan's $f(q)$ which has positive coefficients as well. | |
| Jul 23, 2012 at 20:43 | comment | added | Malik Younsi | +1, I'm convinced..! For my needs, a proof like that suffices (that is, with some estimates and then a numerical argument as in GH's remark), but I still wonder if there's something more going on. More precisely, the following question remains : Is it true that the coefficients in the Taylor series of f are all negative? (this seems to be true, and in particular would imply that all the derivative of $f$ are negative). Anyway, thank you both for this nice argument! | |
| Jul 23, 2012 at 17:37 | comment | added | GH from MO | @juan: I agree, I was typing the same as you! See the "Remark" at the end of my post. Of course the credit is yours. | |
| Jul 23, 2012 at 17:36 | history | edited | GH from MO | CC BY-SA 3.0 |
added 446 characters in body
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| Jul 23, 2012 at 17:36 | comment | added | juan | For $q$ near $1$ this type of reasoning fails, but then we may use the other form of the function. Adding the two arguments all the result may be proved. | |
| Jul 23, 2012 at 16:53 | history | answered | GH from MO | CC BY-SA 3.0 |