
Re: can your CAS help proving inequalities?
Posted:
Mar 9, 2013 10:33 AM


Mate schrieb: > > On Mar 8, 8:08 pm, cliclic...@freenet.de wrote: > > > > Let a, b, c, d be arbitrary real numbers. Define: > > > > 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)) > > > > Can your CAS help proving the following inequalities? > > > > MIN(r(a, b, c, d), r(a, c, b, d), r(a, d, c, b)) <= 0 > > > > MAX(s(a, b, c, d), s(a, c, b, d), s(a, d, c, b)) >= 0 > > > > MIN(t(a, b, c, d), t(a, c, b, d), t(a, d, c, b)) <= 0 > > > > Have fun! > > > > 1. > Denoting > x:=r(a, b, c, d), y:=r(a, c, b, d), z:=r(a, d, c, b) > > ==> x*y+x*z+y*z = 0 > ==> min(x,y,z) <=0 and actually also max(x,y,z) >= 0 > > 2,3. > Denoting similarly x,y,z ==> > > y*z^3+2*y^2*z^2+y^3*z+x*z^3+4*z^2*y*x+4*z*y^2*x+y^3*x > +2*x^2*z^2+4*y*z*x^2+2*x^2*y^2+x^3*z+y*x^3 = 0 > ==> min(x,y,z) <= 0 and also max(x,y,z) >= 0 > > The relations in x,y,z can be easily verified with any CAS. > I have found them using Grobner bases in Maple. > > I had fun indeed. Thanks for the problems. >
Truly formidable  the relations 2,3 in particular! I had omitted the second inequality under 1, because here one already has max(x,y) >= 0, max(x,z) >= 0, max(y,z) >= 0.
Looks like no CAS just simplifies these inequalities to "true".
Martin.

