Mean value problem

Unsolved mathematical problem

In mathematics, the mean value problem was posed by Stephen Smale in 1981.[1] 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} [2]A 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} .[1] 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 has shown 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.[3][4] In 2007, Conte et al. proved that K ≤ 4 d − 1 d + 1 {\displaystyle K\leq 4{\frac {d-1}{d+1}}} ,[2] slightly improving on the bound K ≤ 4 {\displaystyle K\leq 4} for fixed d {\displaystyle d} . In the same year, Crane has shown that K < 4 − 2.263 d {\displaystyle K<4-{\frac {2.263}{\sqrt {d}}}} for d ≥ 8 {\displaystyle d\geq 8} .[5]

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}}}} .[6] The problem of optimizing this lower bound is known as the dual mean value problem.[7]

See also

Notes

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.

References

  1. ^ a b Smale, S. (1981). "The Fundamental Theorem of Algebra and Complexity Theory" (PDF). Bulletin of the American Mathematical Society. New Series. 4 (1): 1–36. doi:10.1090/S0273-0979-1981-14858-8. Retrieved 23 October 2017.
  2. ^ a b Conte, A.; Fujikawa, E.; Lakic, N. (20 June 2007). "Smale's mean value conjecture and the coefficients of univalent functions" (PDF). Proceedings of the American Mathematical Society. 135 (10): 3295–3300. doi:10.1090/S0002-9939-07-08861-2. Retrieved 23 October 2017.
  3. ^ Tischler, D. (1989). "Critical Points and Values of Complex Polynomials". Journal of Complexity. 5 (4): 438–456. doi:10.1016/0885-064X(89)90019-8.
  4. ^ Smale, Steve. "Mathematical Problems for the Next Century" (PDF).
  5. ^ Crane, E. (22 August 2007). "A bound for Smale's mean value conjecture for complex polynomials" (PDF). Bulletin of the London Mathematical Society. 39 (5): 781–791. doi:10.1112/blms/bdm063. S2CID 59416831. Retrieved 23 October 2017.
  6. ^ Dubinin, V.; Sugawa, T. (2009). "Dual mean value problem for complex polynomials". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 85 (9): 135–137. arXiv:0906.4605. Bibcode:2009arXiv0906.4605D. doi:10.3792/pjaa.85.135. S2CID 12020364. Retrieved 23 October 2017.
  7. ^ Ng, T.-W.; Zhang, Y. (2016). "Smale's mean value conjecture for finite Blaschke products". The Journal of Analysis. 24 (2): 331–345. arXiv:1609.00170. Bibcode:2016arXiv160900170N. doi:10.1007/s41478-016-0007-4. S2CID 56272500.