Symmetric functions and the principal case of the Frankl-F\"uredi conjecture

Let $r\geq3$ and $G$ be an $r$-uniform hypergraph with vertex set $\left\{ 1,\ldots,n\right\} $ and edge set $E$. Let \[ \mu\left( G\right) :=\max {\textstyle\sum\limits_{\left\{ i_{1},\ldots,i_{r}\right\} \in E}} x_{i_{1}}\cdots x_{i_{r}}, \] where the maximum is taken over all nonnegative $x_{1},\ldots,x_{n}$ with $x_{1}+\cdots+x_{n}=1.$ Let $t\geq r-1$ be the unique real number such that $\left\vert E\right\vert =\binom{t}{r}$. It is shown that if $r\leq5$ or $t\geq4\left( r-1\right) \left( r-2\right) $, then \[ \mu\left( G\right) \leq t^{-r}\binom{t}{r}% \] with equality holding if and only if $t$ is an integer. The proof is based on some new bounds on elementary symmetric functions.