Polynomial $P_n(z)$ is of degree $n$ and all of its zeros lie in $B(\infty, 1)$. Prove that function $P_n(z) + e^{i\theta}z^n P_n(\frac{1}{\bar{z}})$ only has zeros on $\partial B(0,1)$, $\theta\in\mathbb{R}.$
First we assume $P_n(z) = \Pi_{k=1}^n (z-z_k)$ where $|z_k|>1$ for all $k=1,\dots,n$. I just can't get any information regarding the zeros of $P_n(z) + e^{i\theta}z^n P_n(\frac{1}{\bar{z}})$. Maybe I should apply Maximum Modulus Principle?