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

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

$$$$\begin{split} constant= \frac{4}{3} \end{split}$$$$ is the best $$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!