Mean value problem

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

In 1989, Tischler has shown that the conjecture is true for the optimal bound ${\displaystyle K={\frac {d-1}{d}}}$ if ${\displaystyle f}$ has only real roots, or if all roots of ${\displaystyle f}$ have the same norm.[4][5] In 2007, Conte et al. proved that ${\displaystyle K\leq 4{\frac {d-1}{d+1}}}$,[2] slightly improving on the bound ${\displaystyle K\leq 4}$ for fixed ${\displaystyle d}$. In the same year, Crane has shown that ${\displaystyle K<4-{\frac {2.263}{\sqrt {d}}}}$ for ${\displaystyle d\geq 8}$.[6]

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point ${\displaystyle \zeta }$ such that ${\displaystyle \left|{\frac {f(z)-f(\zeta )}{z-\zeta }}\right|\geq {\frac {|f'(z)|}{n4^{n}}}}$.[7] The problem of optimizing this lower bound is known as the dual mean value problem.[8]

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.