Mean value problem

For the theorem in calculus and real analysis, see Mean value theorem.

The mean value problem (also known as Smale's mean value conjecture) is a problem in the mathematical field of complex analysis first posed by Stephen Smale in 1981.

The problem asks:

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

$\left|\frac{P(z)-P(c)}{z-c}\right|\le K|P^{\prime }(z)|\text{ for }K=1\text{?}$

The question was initially resolved by Smale for $K=4$. For a polynomial of degree $d$, the constant $K$ has to be at least $\frac{d-1}{d}$ due to the example $P(z)=z^d-dz$, so no constant 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, David Tischler showed that the conjecture is true for the optimal bound $K=\frac{d-1}{d}$ if $P$ has only real roots, or if all roots of $P$ have the same norm.

In 2007, Anthony Conte et al. proved that $K\le 4\frac{d-1}{d+1}$, slightly improving on the bound $K\le 4$ for fixed $d$. Also in 2007, Edward Crane showed that $K<4-\frac{2.263}{\sqrt{d}}$ for $d\ge 8$.

Considering the reverse inequality, Vladimir Dubinin and Toshiyuki Sugawa proved that (under the same conditions as above) there exists a critical point $c$ such that $\left|\frac{P(z)-P(c)}{z-c}\right|\ge \frac{|P^{\prime }(z)|}{n{4}^n}$. The problem of optimizing this lower bound is known as the dual mean value problem.

See also

• List of unsolved problems in mathematics
• Köthe conjecture