During my own research, I want to prove the following inequality: $$\frac{f^2r-r+h^2q+\sqrt[]{(f^2r-r+h^2q)^2+4h^2qr}}{2h^2} \geq \frac{{f}^2{r}-{r}-{h}^2{q}+\sqrt[]{({f}^2{r}-{r}-{h}^2{q})^2+4{h}^2{f}^2{q}{r}}}{2{h}^2{f}^2}$$ where ${f}$ and ${h}$ are non-zero real constants, and $ {q},{r}>0$ are real constants.
I tested with some numerical examples whose results show the inequality is correct. I tried to prove it in the past week but without any progress. Can you help me?
Note: In fact, they are the positive roots of the following quadratic equations respectively:
$$ h^2x^2+(r-h^2q-f^2r)x-qr=0 $$ and $$h^2f^2x^2+(r+h^2q-f^2r)x-qr=0$$
after squaring two times and simplifying i got $$16\,{f}^{4}{h}^{4}{q}^{2}r \left( {f}^{4}r+2\,{h}^{2}q{f}^{2}-2\,{f}^{ 2}r+2\,{h}^{2}q+r \right) \geq 0$$ and this is true since $$f^4r-2f^2r+r=r(f^2-1)^2$$