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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 10:17 AM
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On Mar 9, 8:04 pm, "Nasser M. Abbasi" <n...@12000.org> wrote:
> On 3/9/2013 6:07 AM, A N Niel wrote:
>
>
>

> > In Maple, the response
> > {}
> > means there is no solution, while the response

>
> > (that is, no response) means no solution was found.  A third
> > possibility is where some solutions are shown, and then a disclaimer
> > that some solutions may have been lost.

>
> > What CAS did you use, and what does {} mean for it?
>
> Mathematica. It has a special command called
>
> "FindInstance[expr, vars]
>
> finds an instance of vars that makes the statement expr be True.
> gives results in the same form as Solve: if an instance exists,
> and {} if it does not. "
>
> http://reference.wolfram.com/mathematica/ref/FindInstance.html
>
> So that is what I used:
>
> -----------------------------
> Remove["Global`*"]
> r[a_, b_, c_, d_] := (a - c)*(a - d)*(b - c)*(b - d)
> s[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) -
>    Abs[(a - b)*(c - d)]
> t[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) +
>    Abs[(a - b)*(c - d)]
> -----------------------------
>
> and now
>
> ----------------------
> FindInstance[
>   Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
>
> FindInstance[
>   Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
>
> FindInstance[
>   Min[t[a, b, c, d], t[a, c, b, d], t[a, d, c, b]] > 0, {a, b, c, d}]
>
> {}
> -------------------------
>
> But I was not sure this qualifies as "proof" that is why I asked first.
>
> --Nasser



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).







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