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 a complex polynomial ƒ and a complex number z, is there a critical point c of ƒ (i.e. ƒ ′(c) = 0) such that

It was proved for K = 4 (see the article cited above). For a polynomial of degree d the constant K has to be at least (d − 1)/d from the example f(z) = z^d − dz, therefore no bound better than K = 1 can exist. Tischler has some partial results.