The mean value problem is an open problem in the mathematical field of complex analysis first posed by Stephen Smale in 1981. The problem asks:
For a given complex polynomial P {\displaystyle P} of degree d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c} of P {\displaystyle P} (i.e. P ′ ( c ) = 0 {\displaystyle P'(c)=0} ) such that
| P ( z ) − P ( c ) z − c | ≤ K | P ′ ( z ) | for K = 1 ? {\displaystyle \left|{\frac {P(z)-P(c)}{z-c}}\right|\leq K|P'(z)|{\text{ for }}K=1{\text{?}}}
The question has been resolved 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}}} due to the example P ( z ) = z d − d z {\displaystyle P(z)=z^{d}-dz} , so no constant 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, David Tischler showed that the conjecture is true for the optimal bound K = d − 1 d {\displaystyle K={\frac {d-1}{d}}} if P {\displaystyle P} has only real roots, or if all roots of P {\displaystyle P} have the same norm. In 2007, Anthony 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} . Also in 2007, Edward 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, Vladimir Dubinin and Toshiyuki Sugawa proved that (under the same conditions as above) there exists a critical point c {\displaystyle c} such that | P ( z ) − P ( c ) z − c | ≥ | P ′ ( z ) | n 4 n {\displaystyle \left|{\frac {P(z)-P(c)}{z-c}}\right|\geq {\frac {|P'(z)|}{n4^{n}}}} . The problem of optimizing this lower bound is known as the dual mean value problem.
See also List of unsolved problems in mathematics
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