Re: x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 <= C(x1^2 + x2^2 + x3^3 + x4^2)^{3/2}


Kira Yamato wrote:
> I've having trouble showing for any four real numbers x1, x2, x3, x4,
> there exists a real constant C such that
>
> x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 <= C(x1^2 + x2^2 + x3^3 + x4^2)^{3/2}.
>
> This is what I got so far: We can reduce it to the case of
> considering the unit-sphere only. This is because the equation is
> homogeneous. So, next I try to use lagrange multiplier to find the
> maximum value of the left-hand side on the unit-sphere. However, the
> algebra becomes unwielding and I cannot proceed.
>
> Now this was an in-class exam question, and it should not have required
> more than 15 minutes to solve. Am I missing a trick that I don't see?
>
> I also tried using symmetry argument. The left-side side is symmetry
> for every pair of coordinates. So, it's intuitive to guess that the
> maximum occurs at x1=x2=x3=x4. However, I don't know how to justify
> this rigorously.
>
> Please help. Thanks.
>
> -kira

Quick argument:
let y be the largest of x1,x2,x3,x4.
Then x1x2x3, x1x2x4, x1x3x4, and x2x3x4 are all less than or equal to
y^3
The LHS is thus less than 4y^3.
Now, consider the RHS. each square is less than or equal to y^2, so
that side is between Cy^3 and 8Cy^3 (the lower bound is if three of the
x's are 0, the upper bound if they are all equal)
Obviously, if C>=4, the inequality must hold for any x's. Since your
problem only wanted an existance proof, this is enough. If all the x's
are the same, then C=1/2 suffices.

.

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