<s>[INST] Inequality of three variable polynomial I read that one can prove by AM-GM-inequality that for all $a,b,c\in\mathbb{R}_+$ we have that
$$11(a^6 + b^6 + c^6) + 40abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a)$$
How this can be done? Is it possible to prove stronger inequalities like
$$10(a^6 + b^6 + c^6) + 41abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a),$$
or even 
$$a^6 + b^6 + c^6 + 50abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a)$$
without computer?
 [/INST] Each of the inequalities below is easily verified by AM-GM
\begin{eqnarray*}
&29(& a^6 &     &     & + a^3bc^2 & +3a^2b^3c &           &          &          &          & \geq & 5a^3b^2c &          &          &) \\
&29(&     & b^6 &     &           & + a^2b^3c & +3ab^2c^3 &          &          &          & \geq &          & 5ab^3c^2 &          &) \\
&29(&     &     & c^6 & +3a^2b^3c &           & + ab^2c^3 &          &          &          & \geq &          &          & 5a^2bc^3 &) \\
&4 (& a^6 &     &     &           & + a^2b^3c &           &          & +ab^3c^2 &          & \geq & 3a^3b^2c &          &          &) \\
&4 (&     & b^6 &     &           &           & + ab^2c^3 &          &          & +a^2bc^3 & \geq &          & 3ab^3c^2 &          &) \\
&4 (&     &     & c^6 & + a^3bc^2 &           &           & +a^3b^2c &          &          & \geq &          &          & 3a^2bc^3 &). \\
\end{eqnarray*}
Now add these together, cancel some terms, divide by $3$ and we are dumb.
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