Mean value problem

In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks:

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

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

It was proved for ${\displaystyle K=4}$. For a polynomial of degree ${\displaystyle d}$ the constant ${\displaystyle K}$ has to be at least ${\displaystyle {\frac {d-1}{d}}}$ from the example ${\displaystyle f(z)=z^{d}-dz}$, therefore no bound better than ${\displaystyle K=1}$ can exist.