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 $d \ge 2$ and a complex number $z$, is there a critical point $c$ of $f$ such that

:: $\left, \frac \ \le K, f'(z), \textK=1 \text$

It was proved for $K=4$. For a polynomial of degree $d$ the constant $K$ has to be at least $\frac$ from the example $f(z) = z^ - d z$, 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 $K = \frac$ if $f$ has only real roots, or if all roots of $f$ have the same norm. In 2007, Conte et al. proved that $K \le 4 \frac$, slightly improving on the bound $K \le 4$ for fixed $d$. In the same year, Crane has shown that $K < 4-\frac$ for $d \ge 8$.

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point $\zeta$ such that $\left, \frac \ \ge \frac$. 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.

References

{{reflist

1981 introductions
Complex numbers
Unsolved problems in mathematics
Conjectures