Smale's mean value conjecture for complex polynomials

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Smale's mean value conjecture. There exists a critical point $\zeta$ of $P$ such that

$$\left|\frac{P(\zeta)}{\zeta}\right|\le 1-\frac1n.$$

This normalized form is equivalent to Smale's conjecture that the constant $4$ in his mean value estimate can be replaced by $1$. The conjecture remains unresolved in general, although partial estimates are known for several special families of polynomials.