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.

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 showed 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.

In 2007, Conte et al. proved that ${\displaystyle K\leq 4{\frac {d-1}{d+1}}}$, slightly improving on the bound ${\displaystyle K\leq 4}$for fixed ${\displaystyle d}$.

In the same year, Crane showed that ${\displaystyle K<4-{\frac {2.263}{\sqrt {d}}}}$for ${\displaystyle d\geq 8}$.

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 ${\displaystyle \left|{\frac {f(z)-f(\zeta )}{z-\zeta }}\right|\geq {\frac {|f'(z)|}{n4^{n}}}}$.

The problem of optimizing this lower bound is known as the dual mean value problem.

• List of unsolved problems in mathematics