Want to construct a polynomial with specific properties. I'm looking to construct a polynomial $p$ such that the following hold:


*

*$p$ has only one zero in the unit disc.

*$p(1)=1$

*$|p(z)|<1$ for all $|z|\leq 1, z\neq 1$


Context: I was trying to prove that the hypothesis for Rouche's theorem are necessary.  

These properties hold for $p(z) = \frac{1}{5}(z^2 + 3z + 1)$:
It is easy enough to check that it has only one root in the unit circle.  To check that the absolute value condition is met, we can use the maximum principle, and compute $|p(e^{it})|^2 = \frac{1}{25}(2\cos t + 3)^2$ to find that $|p(z)|$ achieves its maximum on the unit disc for $z=1$.
