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Internal cubic symmetric forms in a small number of variables. (English) Zbl 1135.26022
Let $$f$$ be a real symmetric cubic form in $$n$$ variables. The authors wish to characterise the case that $$\mu=\root3\of f$$ is a mean: let us say that $$\mu$$ is internal if $$\min(x_1,\dots,x_n)\leq\mu(x_1,\dots,x_n)\leq\max(x_1,\dots,x_n)$$ for all nonnegative $$x_1,\dots,x_n$$. This note proves that if $$n\leq4$$, then it suffices to test internality for $$x_1,\dots,x_n$$ in $$\{0,1\}$$. The case $$n\geq5$$ is open.

##### MSC:
 26E60 Means 26D05 Inequalities for trigonometric functions and polynomials
##### Keywords:
mean; symmetric cubic form; internality
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