Fejer-Riesz Theorem for analytic polynomials

Question

Let $\mathbb{D}=\{z: |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose $D$ and $E$ are polynomials of degree atmost $n$ with complex coefficients such that

1. $|E(z)| \leq |D(z)|$ for all $z \in \overline{\mathbb{D}}$
2. $D(z) \ne 0$ for all $z \in \mathbb{D}$.
3. $E(z)=z^n\overline{E(1/\overline{z})}$ for all $z \in \mathbb{C}$.
4. $z \mapsto |E(z)|^2-|D(z)|^2$ for $z \in \mathbb{T}$ has only finitely many zeros.

Prove that there exists a polynomial $A$ that does not vanish on $\mathbb{D}$ and $|A(z)|^2=|D(z)|^2-|E(z)|^2$ for all $z \in \overline{\mathbb{D}}$.

My attempt: First note that for $z \in \mathbb{T}$, the function $Q(z)=|E(z)|^2-|D(z)|^2$ is a trigonometric polynomial. By given hypothesis, $Q(z) \geq 0$ on $\mathbb{T}$ and $Q$ is not identically zero. By Fejer-Riesz Theorem, there exists a polynomial $A$ that does not vanish on $\mathbb{D}$ and $|A(z)|^2=|D(z)|^2-|E(z)|^2$ for all $z \in \mathbb{T}$.

How to proceed further? Any hints/ suggestions.