Just an elementary take on the problem, certainly rudimental, but an answer nevertheless. One writes $$ Au + Bu^2 + Cu^3 = u (A+ Bu + Cu^2) $$ and concludes that, in order to fullfill the positiveness condition, the quadratic expression $$ A+ Bu + Cu^2 $$ must not have roots in the interval $[0, 1]$ (an passant, $A> 0$, as it is the value of the polynomial for $u=0$).
If $$ \Delta = B^2 – 4AC < 0$$ the roots are both imaginary, so no problem. If $$ \Delta B^2 – 4AC \geq 0$$ then the following conditions applies
$$\bigg[ \frac{+B + \sqrt {\Delta}}{2C} < 0 \,\,\ OR \,\,\ \frac{+B - \sqrt {\Delta}}{2C} > 1 \bigg] $$ OR $$\bigg[ \frac{+B + \sqrt {\Delta}}{2C} > 1 \,\,\ AND \,\,\ \frac{+B - \sqrt {\Delta}}{2C} < 0 \bigg] $$
(larger root less than zero OR smaller root greater than 1) OR (larger root greater than 1 AND smaller root less than 0)