Mate
Posts:
389
Registered:
8/15/05


Re: can your CAS help proving inequalities?
Posted:
Mar 10, 2013 3:41 PM


On Mar 10, 8:52 pm, "Nasser M. Abbasi" <n...@12000.org> wrote: > 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_{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). > > 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).

