Inequality problem of mathematics If $x,y,z$ are unequal positive quatities then prove that , 
       $(1+x^3)(1+y^3)(1+z^3)>(1+xyz)^3$

lemma: if $a,b,c,x,y,z,p,q,r\ge 0$ then we have
$$(a^3+b^3+c^3)(x^3+y^3+z^3)(p^3+q^3+r^3)\ge (axp+byq+czr)^3$$
poof: note and Use AM-GM inequality
\begin{align*}3&=\sum_{cyc}\dfrac{a^3}{a^3+b^3+c^3}+\sum_{cyc}\dfrac{x^3}{x^3+y^3+z^3}+\sum_{cyc}\dfrac{p^3}{p^3+q^3+r^3}\\
&\ge\sum_{cyc}\dfrac{3axp}{\sqrt[3]{(a^3+b^3+c^3)(x^3+y^3+z^3)(p^3+q^3+r^3)}}\\
\end{align*}
so
$$axp+byq+czr\le \sqrt[3]{(a^3+b^3+c^3)(x^3+y^3+z^3)(p^3+q^3+r^3)}$$
