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 $f$ of degree $d\geq 2$^A and a complex number $z$, is there a critical point $c$ of $f$ (i.e. $f'(c)=0$) such that

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

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

Partial results

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

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

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

See also

• List of unsolved problems in mathematics

Notes

A.^ The constraint on the degree is used but not explicitly stated in Smale (1981); it is made explicit for example in Conte (2007). The constraint is necessary. Without it, the conjecture would be false: The polynomial f(z) = z does not have any critical points.