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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 8, 2013 6:30 PM

On Mar 8, 8:08 pm, cliclic...@freenet.de wrote:
> Hello!
>
> 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!
>
> Martin.

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.
Mate

Date Subject Author
3/8/13 clicliclic@freenet.de
3/8/13 Mate
3/9/13 clicliclic@freenet.de
3/9/13 clicliclic@freenet.de
3/8/13 Nasser Abbasi
3/9/13 Mate
3/9/13 Nasser Abbasi
3/9/13 A N Niel
3/9/13 Mate
3/10/13 A N Niel
3/10/13 Mate
3/9/13 Nasser Abbasi
3/10/13 Mate
3/10/13 Nasser Abbasi
3/10/13 Mate
3/10/13 clicliclic@freenet.de
3/10/13 Nasser Abbasi
3/11/13 Peter Pein
3/11/13 clicliclic@freenet.de
3/11/13 Peter Pein