Timeline for Is there a good reason why $a^{2b} + b^{2a} \le 1$ when $a+b=1$?
Current License: CC BY-SA 2.5
12 events
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| Jan 3, 2013 at 13:32 | comment | added | Shahrooz | Dear Fedja, your solution is nice. Before, I saw the below inequality that is very similar to this one: $$x,y>0, x+y=1 \arrow x^y+y^x\leq sqrt(2)$$. Now, I can see the solution of this inequality. | |
| Nov 27, 2012 at 19:09 | comment | added | jbc | Since I can't edit the above comment, I will have to add a missing bracket in an additional comment. The above estimate holds outside the crucial interval $[\frac 12,1]$, otherwise the bound is $1$. (There are also $2$ end brackets missing). | |
| Nov 27, 2012 at 17:41 | comment | added | jbc | With respect, I would suggest that the following form of the inequality is more interesting for several reasons: $f(a,b)\leq \frac{k^k}{2^{k-1}}$ if $a+b=k$ ($f$ denotes, of course, the expression to be maximised It includes the cases $a+b=\frac 12$ and $a+b=1$, is more precise and displays more interesting behaviour (the dependence of the maximum on $k$ and the nature of the point where it is assumed. | |
| Apr 13, 2010 at 15:22 | comment | added | Sunni | @fedja: That is good. For me, I learned a good proof, that is all I want for posting this problem. | |
| Apr 13, 2010 at 14:26 | comment | added | fedja | Thanks. Well, anybody who is interested can find it here, so I do not think it makes much sense to submit it anywhere else. I'll, probably, just send a PM to Vasile. | |
| Apr 13, 2010 at 14:22 | history | edited | fedja | CC BY-SA 2.5 |
added the bottom line proof
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| Apr 13, 2010 at 13:26 | comment | added | Sunni | @fedja: I think your proof meets the criteria. If you are interested, you may submit your proof featuring solving a conjecture in <On Some Inequalities With Power-Exponential Functions> by Vasile Cirtoaje. | |
| Apr 13, 2010 at 13:21 | vote | accept | Sunni | ||
| Apr 13, 2010 at 4:26 | comment | added | Tom Leinster | Bravo ! | |
| Apr 13, 2010 at 3:50 | history | edited | fedja | CC BY-SA 2.5 |
fixed the proof
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| Apr 12, 2010 at 17:28 | history | edited | fedja | CC BY-SA 2.5 |
deleted 107 characters in body
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| Apr 12, 2010 at 17:17 | history | answered | fedja | CC BY-SA 2.5 |