Math Forum / Mathematics / General Topics / October 2008

Let's consider real x,y,z,... from  range (0,infinity). We are
interested in true inequalities homogeneous of n-th order in variables
x,y,z, ... The problem is to find all such inequalities.

1) for n=1, we have:
        A*(x+y)>=0, so the  constant A is from S(A)={A: A>=0}

2) for n=2, we have:
       A*(x^2+y^2)+B*x*y>=0, it's easy to show that constants A,B
have to satisfy inequality
       abs(B)>2*abs(A), so S(A,B)={(A,B): abs(A)>2*abs(B)}, and this
set is a 2D cone with origin at (0,0)
3) for n=3, we have
       A*(x^3+y^3+z^3)+B*(x^2y+y^*x+...)+C*xyz>=0,
I suppose that S(A,B,C) is also a cone with origin at (0,0,0), but I
have no idea how to prove it
4) for n=4, we have:
  A(x^4+y^4+z^4+t^4)+B(xyz^2+...)+C(x^2y^2+...)+D(x^3y+...)+Exyzt>=0
I'm interesting if it's also a cone S(A,B,C,D,E) ...

and if it's a general rule. Are there some simply methods for finding
such sets?