Timeline for Arithmetic expansion of harmonic sum
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2020 at 5:03 | history | edited | Yoshitaka | CC BY-SA 4.0 |
added 59 characters in body
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| Jul 10, 2020 at 4:41 | comment | added | Yoshitaka | I am sorry for the inconvenience to change the initial query (whose solution must be collect with yours). I was happy to have your helpful comment on the problem which troubles me. | |
| Jul 10, 2020 at 4:41 | comment | added | Yoshitaka | It should be better to modify my question as follows. Suppose that $H$ is written as $\frac{1}{\frac{w_1}{x_1} + \cdots + \frac{w_d}{x_d}}$. Then, is it possible to express $H$ in an arithmetic form like $H \geq \Sigma_i (\tau_i(w_1, \ldots, w_d)\cdot \theta_i(x_1, \ldots, x_d))$ with equality $w_k = x_k (1\leq k \leq d)$, where $\tau_i$ (resp. $\theta_i$) is any function independent from the variables $x_1, \ldots, x_d$ (resp. the weights $w_1,\ldots, w_d$)? | |
| Jul 10, 2020 at 4:18 | comment | added | Yoshitaka | Thank you very much. That's very helpful for me. And now, I notice that I should have written the equality condition as $w_k = \frac{1}{x_k}$, not $w_k = x_k$. I am sorry for that. My concern is that though the Schwarz inequality holds, when we consider $w_k = \frac{1}{x_k}$ as the expected equality condition, $H$ does not reach the right term; $\frac{1}{\sqrt{\Sigma_i w_i^2}}\frac{1}{\sqrt{\Sigma_i x_i^2}}$. | |
| Jul 10, 2020 at 2:58 | comment | added | fedja | What's wrong with $H\ge \frac 1{\sqrt{ \sum_i w_i^2}} \frac 1{\sqrt {\sum_i x_i^2}}$? | |
| Jul 8, 2020 at 3:20 | review | First posts | |||
| Jul 8, 2020 at 5:52 | |||||
| Jul 8, 2020 at 3:18 | history | asked | Yoshitaka | CC BY-SA 4.0 |