Condition for zeros of a polynomial in Unit Disk Consider a polynomial in $\mathbb{C}$ with complex coefficients, $\lambda^2+p\lambda+q$ where both $p$ and $q$ are complex numbers. 

I am looking a for a condition of $p$ and $q$ such that the zeros of this polynomial would lie in the unit open disk. In other words the modulus of the zeros is strictly less than $1$.
If $p\in\mathbb{R}$ then the roots of $z^2+p z+q$ lie in the open unit disc if and only if $$ |p| &lt; \frac{1-|q|^2}{|1-q|}.$$ 

Note that for $q\in\mathbb{R}$ this criterion becomes $|q|&lt;1$ and $|p|&lt;1+q$.  If $p=|p|\omega$ for some $\omega$ on the unit circle then the criterion for real $p$ can be extended to
$$|p| &lt; \frac{1-|q|^2}{|\omega^2-q|}$$
which is valid also for complex $p$.

The denominator is bounded above by $1+|q|$.  In other words $|p|+|q|&lt;1$ will always imply that the roots will lie in the open unit disc.  This latter bound also guarantees that $z^2$ and $z^2+pz+q$ have the same number of roots in the open unit disc by Rouche's theorem.
I found the answer in the fantastic article written by Rahaman et.al. On the Eneström-Kakeya theorem. Tohoku Mathematical Journal, $Vol. 20 (2)$, $(1968)$, $126-136$.

Thanks to Q. I. Rahaman.