Seeking the Maximum of a Product expression using Inequalities

Question

Since $\ln a\leq a-1,$ I tried to use that inequality to calculate a maximum value for $$\prod_{cyc}\frac{9y+4z-6x}{x}$$ where $x,y,z>0$. Or $$\sum_{cyc}\log\frac{9y+4z-6x}{x}$$ Then $$\log\frac{9y+4z-6x}{x}\leq\frac{9y+4z-7x}{x}$$ but it seems not to go further.

Answer 1

For $x=y=z$ we get a value $343$.

We'll prove that it's a maximal value.

Indeed, we need to prove that $$\prod_{cyc}\frac{9y+4z-6x}{x}\leq343$$ or $$\sum_{cyc}(36x^3-92x^2y+33x^2z+23xyz)\geq0.$$ Now, let $x=\min\{x,y,z\}$, $y=x+u$ and $z=x+v$.

Thus, we need to prove that $$49(u^2-uv+v^2)x+(9u+4v)(2u-3v)^2\geq0,$$ which is obvious.

Done!