Is there any complex polynomial $p$ of one variable having no zeros within the unit square $-1 < \Re(z) , \Im(z) < 1$ such that $\left|p(0)\right|$ is strictly smaller than $\left|p(z)\right|$ whenever $z$ is either on the corner of the square or on the middle of one the square's sides? That is, when $z \in \{1, 1+i, i, -1+i, -1, -1-i, -i, 1-i\}$.

This problem may be quite arbitrary, so any theorems are welcome where some finite number of inequalities (where on each side of each inequality the only relevant parameters are evaluations of the polynomial in a finite number of points) implies a strong bound on some zero of the polynomial.