In mathematics, the mean value problem was posed by Stephen Smale in 1981.
[1] This problem is still open in full generality.
The problem asks: It was proved for
[1] For a polynomial of degree
, therefore no bound better than
The conjecture is known to hold in special cases; for other cases, the bound on
could be improved depending on the degree
, although no absolute bound
is known that holds for all
In 1989, Tischler showed that the conjecture is true for the optimal bound
has only real roots, or if all roots of
[3][4] In 2007, Conte et al. proved that
,[2] slightly improving on the bound
In the same year, Crane showed that
[5] Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point
ζ
f ( z ) − f ( ζ )
[6] The problem of optimizing this lower bound is known as the dual mean value problem.