It was
proved for .[1] For a polynomial of degree the constant has to be at least from the example , therefore no bound better than can exist.
Partial results
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 has shown that the conjecture is true for the optimal bound if has only
realroots, or if all roots of have the same
norm.[3][4] In 2007, Conte et al. proved that ,[2] slightly improving on the bound for fixed . In the same year, Crane has shown that for .[5]
Considering the reverse
inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point such that .[6] The problem of optimizing this lower bound is known as the
dual mean value problem.[7]
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.
It was
proved for .[1] For a polynomial of degree the constant has to be at least from the example , therefore no bound better than can exist.
Partial results
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 has shown that the conjecture is true for the optimal bound if has only
realroots, or if all roots of have the same
norm.[3][4] In 2007, Conte et al. proved that ,[2] slightly improving on the bound for fixed . In the same year, Crane has shown that for .[5]
Considering the reverse
inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point such that .[6] The problem of optimizing this lower bound is known as the
dual mean value problem.[7]
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.