Mate
Posts:
389
Registered:
8/15/05


Re: can your CAS help proving inequalities?
Posted:
Mar 10, 2013 10:17 AM


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_{n1}/(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).

