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 f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z}, is there a critical point c {\displaystyle c} of f {\displaystyle f} (i.e. f ′ ( c ) = 0 {\displaystyle f'(c)=0}) such that

| f ( z ) − f ( c ) z − c | ≤ K | f ′ ( z ) | for K = 1 ? {\displaystyle \left|{\frac {f(z)-f(c)}{z-c}}\right|\leq K|f'(z)|{\text{ for }}K=1{\text{?}}}

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

Partial results

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

In 1989, Tischler showed that the conjecture is true for the optimal bound K = d − 1 d {\displaystyle K={\frac {d-1}{d}}} if f {\displaystyle f} has only real roots, or if all roots of f {\displaystyle f} have the same norm.

In 2007, Conte et al. proved that K ≤ 4 d − 1 d + 1 {\displaystyle K\leq 4{\frac {d-1}{d+1}}}, slightly improving on the bound K ≤ 4 {\displaystyle K\leq 4} for fixed d {\displaystyle d}.

In the same year, Crane showed that K < 4 − 2.263 d {\displaystyle K<4-{\frac {2.263}{\sqrt {d}}}} for d ≥ 8 {\displaystyle d\geq 8}.

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point ζ {\displaystyle \zeta } such that | f ( z ) − f ( ζ ) z − ζ | ≥ | f ′ ( z ) | n 4 n {\displaystyle \left|{\frac {f(z)-f(\zeta )}{z-\zeta }}\right|\geq {\frac {|f'(z)|}{n4^{n}}}}.

The problem of optimizing this lower bound is known as the dual mean value problem.

See also

Notes