I’m interested by the following problem :

Let $ x,y,z>0$ with $ 49x-7y+z>0$ , $ 49y-7z+x>0$ , $ 49z-7x+y>0$ then we have : $ $ \Big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\Big)\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{43^3}}\leq \sqrt{3(x+y+z)}$ $

I have tested this inequality with Pari-Gp and it seems to be okay . Furthermore I think we can use the $ uvw$ ‘s method (because the equality case comes when $ x=y=z$ ) but I don’t see how now . I have a ugly proof using derivative (the inequality can be reduce to a two variable inequality) but it’s too long to be explain here . The inequality is too precise to use Jensen’s or Slater’s inequality here. Finally I have also tested brut force but I can’t find an interesting irreductible factorization .

If you have a hint it would be nice .

Thanks in advance .