A Fibonacci Inequality

Question

While attempting to seek new inequalities for Fibonacci numbers one was developed, empirically, but has a tighter bound than several comparable inequalities provide (Titu's, Cauchy's, etc). The inequality developed is: $$ \frac{F_{n}^{6}}{F_{n+1}^{3} + F_{n+2}^{3}} + \frac{F_{n+1}^{6}}{F_{n}^{3} + F_{n+2}^{3}} + \frac{F_{n+2}^{6}}{F_{n}^{3} + F_{n+1}^{3}} \geq \frac{27}{2} \, F_{n} \, F_{n+1} \, F_{n+2}. $$ What is being asked for is assistance in finding some form of known inequality that will help prove this inequality presented.

Answer 1

Asymptotically, $$ F_n \approx \dfrac{\varphi^n}{\sqrt{5}}, \quad\mbox{ where }\;\varphi = \dfrac{1+\sqrt{5}}{2}.$$

Then we have

$$ LHS \approx \dfrac{\varphi^{3n}}{\sqrt{5}^3} \left(\dfrac{1}{\varphi^{3}+\varphi^{6}} + \dfrac{\varphi^{6}}{1+\varphi^{6}} + \dfrac{\varphi^{12}}{1+\varphi^{3}} \right) \approx 5.589 \varphi^{3n}. $$

$$ RHS \approx\dfrac{\varphi^{3n}}{\sqrt{5}^3} \cdot \dfrac{27\varphi^3}{2} \approx 5.115 \varphi^{3n}. $$

Playing with small $n$, one can improve constant $\dfrac{27}{2}$ to $\dfrac{486}{35}$ (then we'll have equality for $n=2$).