Mean value problem

The mean value problem is an open problem in the mathematical field of complex analysis first posed by Stephen Smale in 1981. The problem asks:

For a given complex polynomial ${\displaystyle P}$ of degree ${\displaystyle d\geq 2}$ and a complex number ${\displaystyle z}$, is there a critical point ${\displaystyle c}$ of ${\displaystyle P}$ (i.e. ${\displaystyle P'(c)=0}$) such that

${\displaystyle \left|{\frac {P(z)-P(c)}{z-c}}\right|\leq K|P'(z)|{\text{ for }}K=1{\text{?}}}$

The question has been resolved for ${\displaystyle K=4}$. For a polynomial of degree ${\displaystyle d}$, the constant ${\displaystyle K}$ has to be at least ${\displaystyle {\frac {d-1}{d}}}$ due to the example ${\displaystyle P(z)=z^{d}-dz}$, so no constant bound better than ${\displaystyle K=1}$ can exist.