// Workers AI · dad joke modeWhat did the mean value problem say? I'm averaging an attitude.

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

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

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 $K\leq 4{\frac {d-1}{d+1}}$, slightly improving on the bound $K\leq 4$ for fixed $d$. Also in 2007, Edward Crane showed that $K<4-{\frac {2.263}{\sqrt {d}}}$ for $d\geq 8$.

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 $\left|{\frac {P(z)-P(c)}{z-c}}\right|\geq {\frac {|P'(z)|}{n4^{n}}}$. The problem of optimizing this lower bound is known as the dual mean value problem.

See also

• List of unsolved problems in mathematics
• Köthe conjecture