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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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Mate

Posts: 389
Registered: 8/15/05
Re: can your CAS help proving inequalities?
Posted: Mar 10, 2013 3:41 PM
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On Mar 10, 8:52 pm, "Nasser M. Abbasi" <n...@12000.org> wrote:
> On 3/10/2013 8:17 AM, Mate wrote:
>
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> > 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



What do you mean by solutions? Probably you mean the MIN (or INF) of
the LHS.
The inequality is either true (for all x_i > 0)
or a counterexample must be found. For n < 14 it is true. For n=14 the
counterexample
is contained in the wiki article you cited.

Actually I am almost sure that Mathematica cannot be used for these,
except maybe for counterexamples but indeed very time consuming.
(But I am a Maple user, I do not use Mathematica).




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