Shalom aleichem.
Excuse me ... .
I try to simply this problem ... .
If
\[ x=\ln{a}, \]
\[ y=\ln{b}, \]
\[ z=\ln{c}, \]
this problem become:
\[ (y/x)^4+(z/y)^4+(x/z)^4\geq{y/x}+{z/y}+{x/z} \]
if $x,y,z\in(-\infty,0)$.
Thank you ... .
Terpujilah Kristus.