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Topic: can your CAS help proving inequalities?
Replies: 19   Last Post: Mar 11, 2013 12:00 PM

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Nasser Abbasi

Posts: 5,688
Registered: 2/7/05
Re: can your CAS help proving inequalities?
Posted: Mar 10, 2013 2:52 PM
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On 3/10/2013 8:17 AM, Mate wrote:

>
> There are many simple but tough inequalities,
> e.g. the cyclic inequalities (Shapiro):
>
> x_1/(x_2+x_3) + x_2/(x_3+x_4) + ... + x_{n-1}/(x_n + x_1) + x_n/(x_1 +
> x_2) >= n/2
>
> for x_i > 0.
>
> It would be interesting to know if Mathematica can manage these.
>
> So, what is Mathematica's answer for n in {6, 8, 10, 11, 14, 15}.
>
> (for n=14 there exists a counterexample, for n=15 the answer seems to
> be not known).
>


I am running it now for even n. But it is very time consuming,
still waiting for n=6. For n=2, n=4 these are the results found so
far:

{2, {{xx[1] -> 1, xx[2] -> 1}},

{4, {{xx[1] -> 1, xx[2] -> 1/2, xx[3] -> 1, xx[4] -> 1/4}}}}

{6, ..... will check in few hours ....}

Are these solutions listed somewhere? I searched but did not find them.

http://en.wikipedia.org/wiki/Shapiro_inequality

--Nasser





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