I have finally found the right function such that we have :
Let $a,b,c>0$ such that $abc=a+b+c$ then we have : $$\sum_{cyc}\tan\Big(\frac{1}{7a+b}\Big)\leq 3\tan\Big(\frac{1}{8\sqrt{3}}\Big)$$
To underline the difficulty we have also :
Let $a,b,c>0$ such that $\pi=a+b+c$ and $a,b,c<\frac{\pi}{2}$then we have : $$\sum_{cyc}\tan\Big(\frac{1}{7\tan(a)+\tan(b)}\Big)\leq 3\tan\Big(\frac{1}{8\sqrt{3}}\Big)$$
It's more impressive like this and I don't know what to do with this in fact it's more a present for Michael Rozenberg .
I accept ugly proof but I think it's too hard to find .
Any hint would be appreciable .
Thanks in advance .