This is not an answer, this is a message to @Gjergji Zaimi.
Thank You very much. Your answer is true with this version. Please see comment of me and You; me and I Wolfgang. I am sorry, because now I see detail again. I think this inequality is not fomulate true with my ideas, so maybe I must re-fomulate my ideas again. And I need your help for symbol. See example:
Inequality 1 with $n=3$, $m=2$.
$${x_1}^{a_1}.{x_2}^{a_2}+{x_1}^{a_1}.{x_3}^{a_2}+{x_2}^{a_1}.{x_3}^{a_2}+{x_1}^{a_2}.{x_2}^{a_3}+{x_1}^{a_2}.{x_3}^{a_3}+{x_2}^{a_2}.{x_3}^{a_3}+{x_1}^{a_1}.{x_2}^{a_3}+{x_1}^{a_1}.{x_3}^{a_3}+{x_2}^{a_1}.{x_3}^{a_3}+{x_1}^{a_2}.{x_2}^{a_1}+{x_1}^{a_2}.{x_3}^{a_1}+{x_2}^{a_2}.{x_3}^{a_1}+{x_1}^{a_3}.{x_2}^{a_2}+{x_1}^{a_3}.{x_3}^{a_2}+{x_2}^{a_3}.{x_3}^{a_2}+{x_1}^{a_3}.{x_2}^{a_1}+{x_1}^{a_3}.{x_3}^{a_1}+{x_2}^{a_3}.{x_3}^{a_1} \leq {y_1}^{a_1}.{y_2}^{a_2}+{y_1}^{a_1}.{y_3}^{a_2}+{y_2}^{a_1}.{y_3}^{a_2}+{y_1}^{a_2}.{y_2}^{a_3}+{y_1}^{a_2}.{y_3}^{a_3}+{y_2}^{a_2}.{y_3}^{a_3}+{y_1}^{a_1}.{y_2}^{a_3}+{y_1}^{a_1}.{y_3}^{a_3}+{y_2}^{a_1}.{y_3}^{a_3}+{y_1}^{a_2}.{y_2}^{a_1}+{y_1}^{a_2}.{y_3}^{a_1}+{y_2}^{a_2}.{y_3}^{a_1}+{y_1}^{a_3}.{y_2}^{a_2}+{y_1}^{a_3}.{y_3}^{a_2}+{y_2}^{a_3}.{y_3}^{a_2}+{y_1}^{a_3}.{y_2}^{a_1}+{y_1}^{a_3}.{y_3}^{a_1}+{y_2}^{a_3}.{y_3}^{a_1} $$
Inequality 2 with $n=3$, $m=2$.
$${a_1}^{x_1}.{a_2}^{x_2}+{a_1}^{x_1}.{a_3}^{x_2}+{a_2}^{x_1}.{a_3}^{x_2}+{a_1}^{x_2}.{a_2}^{x_3}+{a_1}^{x_2}.{a_3}^{x_3}+{a_2}^{x_2}.{a_3}^{x_3}+{a_1}^{x_1}.{a_2}^{x_3}+{a_1}^{x_1}.{a_3}^{x_3}+{a_2}^{x_1}.{a_3}^{x_3}+{a_1}^{x_2}.{a_2}^{x_1}+{a_1}^{x_2}.{a_3}^{x_1}+{a_2}^{x_2}.{a_3}^{x_1}+{a_1}^{x_3}.{a_2}^{x_2}+{a_1}^{x_3}.{a_3}^{x_2}+{a_2}^{x_3}.{a_3}^{x_2}+{a_1}^{x_3}.{a_2}^{x_1}+{a_1}^{x_3}.{a_3}^{x_1}+{a_2}^{x_3}.{a_3}^{x_1} \geq {a_1}^{y_1}.{a_2}^{y_2}+{a_1}^{y_1}.{a_3}^{y_2}+{a_2}^{y_1}.{a_3}^{y_2}+{a_1}^{y_2}.{a_2}^{y_3}+{a_1}^{y_2}.{a_3}^{y_3}+{a_2}^{y_2}.{a_3}^{y_3}+{a_1}^{y_1}.{a_2}^{y_3}+{a_1}^{y_1}.{a_3}^{y_3}+{a_2}^{y_1}.{a_3}^{y_3}+{a_1}^{y_2}.{a_2}^{y_1}+{a_1}^{y_2}.{a_3}^{y_1}+{a_2}^{y_2}.{a_3}^{y_1}+{a_1}^{y_3}.{a_2}^{y_2}+{a_1}^{y_3}.{a_3}^{y_2}+{a_2}^{y_3}.{a_3}^{y_2}+{a_1}^{y_3}.{a_2}^{y_1}+{a_1}^{y_3}.{a_3}^{y_1}+{a_2}^{y_3}.{a_3}^{y_1} $$
Can You help me reformulate in general case: