Mean value problem

Short description: Unsolved mathematical 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 [math]\displaystyle{ f }[/math] of degree [math]\displaystyle{ d \ge 2 }[/math] and a complex number [math]\displaystyle{ z }[/math], is there a critical point [math]\displaystyle{ c }[/math] of [math]\displaystyle{ f }[/math] (i.e. [math]\displaystyle{ f'(c) = 0 }[/math]) such that

[math]\displaystyle{ \left| \frac{f(z) - f(c)}{z - c} \right| \le K|f'(z)| \text{ for }K=1 \text{?} }[/math]

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

Partial results

The conjecture is known to hold in special cases; for other cases, the bound on [math]\displaystyle{ K }[/math] could be improved depending on the degree [math]\displaystyle{ d }[/math], although no absolute bound [math]\displaystyle{ K\lt 4 }[/math] is known that holds for all [math]\displaystyle{ d }[/math].

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

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point [math]\displaystyle{ \zeta }[/math] such that [math]\displaystyle{ \left| \frac{f(z) - f(\zeta)}{z - \zeta} \right| \ge \frac{|f'(z)|}{n 4^{n}} }[/math]. 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.