Let $$ D = x_1x_2...x_{m-1} + x_1x_2...x_{m-2} + ... x_1 + 1$$ $$ S = x_1 x_2...x_m - 1 > 0 $$ and $$x_i > 0, i=1,...,m$$ For inequalities $S-D$ and its cyclic-symmetric derivatives: $$ F(x_1, x_2, ..., x_m) = x_1x_2...x_m - x_1x_2...x_{m-1} - x_1x_2...x_{m-2} - ... x_1 - 2 < 0 \\ F(x_m, x_1, x_2, ..., x_{m-1}) = x_mx_1x_2...x_{m-1} - x_mx_1x_2...x_{m-2} - x_mx_1x_2...x_{m-3} - ... x_m - 2 < 0 \\ ... \\ F(x_2, ..., x_m, x_1) = x_2...x_mx_1 - x_2...x_{m} - x_2...x_{m-1} - ... x_2 - 2 < 0 $$ The set of its solution is: $$M = \left\{ (x_1, ..., x_m)| F(x_1, ... , x_m) < 0, F(x_m, x_1, ...x_{m-1}) < 0, ..., F(x_2, ..., x_m, x_1) < 0 \right\} $$
Frankly speaking, I don't know how to solve the inequality $F < 0$, but here are some following properties that I found in doing so.
if point $P_1 = (c_1, c_2, ..., c_m) \in M$ then points $P_2 = (c_2,c_3,..., c_m, c_1), P_3, ... P_m = (c_m, c_1, ..., c_{m-1})$ are also in set $M$.
Set $M'=\left\{ (x_1,...,x_m) | 0 < x_i < 2 , \prod_{i=1}^m x_i > 1 \right\} \subseteq M$
Since $F(x_1, ..., x_m) = x_1...x_{m-1}(x_m-2) + x_1...x_{m-2}(x_{m-1}-2) + ... x_1(x_2 -2) + (x_1 -2) < 0$ and $x_i > 0,$
Now my question is
- Are there any new property or any books/papers related to $F$ (or $M$) ?
- For $(x_1, x_2, ..., x_m) \in M$, $x_i$ has the same range(according to property 1), Does there exists $b>0$ such that $x_i<b, i=1,...,m$ ($x_i$ is bounded)