I can present a solution here.Is it correct?
Multiplying both sides with $a^{4}b^{4}c^{4}$ the inequality becomes
$\displaystyle\sum_{sym}a^{4}b^{4}c^{3}\leq \displaystyle\sum_{sym}a^{9}bc$.
According to Muirhead's theorem, it suffices to prove that $6[4,4,3]\leq 6[9,1,1]$ which is true by Muirhead if I am not mistaken.Am I correct?