Timeline for How to prove the monotonicity of the following function? $f(x) = \frac{\sum\limits_{i=1}^K\ln(1+a_ix)}{\sum\limits_{i=1}^K\ln(1+\frac{1}{2}a_ix)}$
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| Sep 14, 2016 at 16:06 | answer | added | fedja | timeline score: 2 | |
| Sep 14, 2016 at 9:15 | comment | added | mingzhanzhang | @fedja: When $x$ is very large so that $a_ix \gg 2$, then the ``1'' in the logarithm can be ignored and the statement can be proved. Besides, the simulation Matlab codes are below, I have tried many times and still cann't find the counterexample, Could you present some examples? K = 10; m = 0; n = 100; a = m + (n-m) .* rand(K,1); x = [0.001:0.01:10]; f1 = sum( log(1 + ax ), 1 ); f2 = sum( log(1 + ax/2), 1 ); f = f1./f2; figure plot(x, f, '-r') | |
| Sep 14, 2016 at 6:10 | comment | added | fedja | There is no way. The statement is false in general (which makes me extremely curious about what you mean by the "proof by simple simulation": I hope you haven't just considered a finite number of random sets of parameters) | |
| Sep 13, 2016 at 0:59 | comment | added | Brendan McKay | I don't know if OPs are pinged when comments are added to solutions, so I'm adding this to alert mingzhanzhang. | |
| Sep 12, 2016 at 15:05 | vote | accept | mingzhanzhang | ||
| Sep 13, 2016 at 9:15 | |||||
| Sep 12, 2016 at 10:20 | review | First posts | |||
| Sep 12, 2016 at 10:32 | |||||
| Sep 12, 2016 at 10:18 | history | asked | mingzhanzhang | CC BY-SA 3.0 |