On a nonlinear differential subordination I

We find conditions on the complex-valued functions $A,B,C,D$ in the unit disc $U$ such that the differential inequality
$$ |A(z)z^2 p''(z)+B(z)p^2(z)+C(z)p(z)+D(z)|<M $$
implies $|p(z)|<N$, where $p$ is analytic in $U$, with $p(0)=0$.