In mathematics, the mean value problem was posed by Stephen Smale in 1981.^[1] This problem is still open in full generality. The problem asks:

For a given complex polynomial ${\displaystyle f}$ of degree ${\displaystyle d\geq 2}$^[2]A 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 ${\displaystyle K=4}$.^[1] 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.

Partial results

The conjecture is known to hold in special cases; for other cases, the bound on ${\displaystyle K}$ could be improved depending on the degree ${\displaystyle d}$, although no absolute bound ${\displaystyle K<4}$ is known that holds for all ${\displaystyle d}$.

In 1989, Tischler has shown that the conjecture is true for the optimal bound ${\displaystyle K={\frac {d-1}{d))}$ if ${\displaystyle f}$ has only real roots, or if all roots of ${\displaystyle f}$ have the same norm.^[3][4] In 2007, Conte et al. proved that ${\displaystyle K\leq 4{\frac {d-1}{d+1))}$,^[2] slightly improving on the bound ${\displaystyle K\leq 4}$ for fixed ${\displaystyle d}$. In the same year, Crane has shown that ${\displaystyle K<4-{\frac {2.263}{\sqrt {d))))$ for ${\displaystyle d\geq 8}$.^[5]

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point ${\displaystyle \zeta }$ such that ${\displaystyle \left|{\frac {f(z)-f(\zeta )}{z-\zeta ))\right|\geq {\frac {|f'(z)|}{n4^{n))))$.^[6] The problem of optimizing this lower bound is known as the dual mean value problem.^[7]