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
6 events
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| Mar 5, 2010 at 18:48 | comment | added | David E Speyer | @GZ: If I were the referee, I would insist on the author distinguishing results that don't have proofs from results that don't have elementary proofs. But in any case, nice work finding that reference! That definitely makes it seem less likely that there is an elegant approach. | |
| Mar 5, 2010 at 18:41 | comment | added | Gjergji Zaimi | @DS: I'm sure that's what the author meant by stating it as a "conjecture", that there is no solution similar to the simple proofs of the other inequalities mentioned in the article. I just gave it as a counterargument to miwalin insisting that we find an elegant solution here. | |
| Mar 5, 2010 at 18:39 | comment | added | David E Speyer | Wait, this was actually published as a conjecture? I'm certain this inequality is provable if you let me use a computer. For example, let $f[t]=t^{2(1−t)}$. Then, for $k/10000 \leq t \leq (k+1)/10000$ , we have $f[t]+f[1−t] \leq f[(k+1)/10000]+f[1−k/10000]$ . This proves the inequality for all $t$ not in $[0,3/10000]$ , $[4747/10000, 5253/10000]$ and $[9997/10000,1] $. Local arguments near 0, $1/2$ and 1 should finish the job. This is only a hard problem if you insist on a simple, non-machine-aided answer. | |
| Mar 5, 2010 at 18:30 | comment | added | gowers | An argument that backs this up to some extent is the fact that the maximum occurs at (0,1), (1/2,1/2) and (1,0). What's more, the place where the minimum occurs is, if my calculation is correct, the place where aloga = 1-a, which doesn't fill one with confidence that a slick solution exists. Even so, I don't completely rule it out. | |
| Mar 5, 2010 at 18:26 | comment | added | Sunni | It now becomes a problem whether I should post this sort of problem here (my interest in this problem is simple: curiosity). Yes, generally problems from olympiad-style is not welcome here. I expect that new and fresh ideas may appear here. | |
| Mar 5, 2010 at 18:15 | history | answered | Gjergji Zaimi | CC BY-SA 2.5 |