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

In article <pan.2005.08.17.22.38.00.379490@xxxxxxxxxxxxx> Kira Yamato <kirakun@xxxxxxxxxxxxx> writes:
>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?
>
Hint: Find a constant C1 such that

x2 x3 x4 <= C1 (x1^2 + x2^2 + x3^2 + x4^2)(3/2).

Add four such inequalities.
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