Prove \[\begin{equation}\begin{split} \prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 \end{split}\end{equation}\]

Prove $ $ \begin{equation}\begin{split} \prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 \end{split}\end{equation}$ $ with $ a,\,b,\,c> 0$ .

$ $ \begin{equation}\begin{split} constant= \frac{4}{3} \end{split}\end{equation}$ $ is the best $ constant$ , which was found by me (using discriminant and uvw).

I can’t use Titu lemma and Holder inequality here, but without success, so I need helps. Thanks!