Mean value 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 ${\displaystyle f}$ of degree ${\displaystyle d\geq 2}$^[2]A and a complex number ${\displaystyle z}$, is there a critical point ${\displaystyle c}$ of ${\displaystyle f}$ (i.e. ${\displaystyle f'(c)=0}$) such that

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

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

Partial results

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 showed 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.^[3][4]

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 showed that ${\displaystyle K<4-{\frac {2.263}{\sqrt {d}}}}$ for ${\displaystyle d\geq 8}$.^[5]

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point ${\displaystyle \zeta }$ such that ${\displaystyle \left|{\frac {f(z)-f(\zeta )}{z-\zeta }}\right|\geq {\frac {|f'(z)|}{n4^{n}}}}$.^[6]

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

See also

• List of unsolved problems in mathematics

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. 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. 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.