Mean value problem

The mean value problem (also known as Smale's mean value conjecture) is a problem in the mathematical field of complex analysis first posed by Stephen Smale in 1981.

The problem asks:

For a given complex polynomial

P

of degree

d\geq 2

and a complex number

z

, is there a critical point

c

of

P

(i.e.

P'(c)=0

) such that

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

The question was initially resolved by Smale for $K=4$ . For a polynomial of degree $d$ , the constant $K$ has to be at least ${\frac {d-1}{d}}$ due to the example $P(z)=z^{d}-dz$ , so no constant bound better than $K=1$ can exist.

Partial results

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

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

In 2007, Anthony Conte et al. proved that $K\leq 4{\frac {d-1}{d+1}}$ , slightly improving on the bound $K\leq 4$ for fixed $d$ . Also in 2007, Edward Crane showed that $K<4-{\frac {2.263}{\sqrt {d}}}$ for $d\geq 8$ .

Considering the reverse inequality, Vladimir Dubinin and Toshiyuki Sugawa proved that (under the same conditions as above) there exists a critical point $c$ such that $\left|{\frac {P(z)-P(c)}{z-c}}\right|\geq {\frac {|P'(z)|}{n4^{n}}}$ . The problem of optimizing this lower bound is known as the dual mean value problem.

Notes

↑ 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 P(z) = z does not have any critical points.

References

1 2 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.
1 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.
↑ 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.
↑ Smale, Steve. "Mathematical Problems for the Next Century" (PDF).
↑ 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.
↑ 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.
↑ 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.

[3] 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 P(z) = z does not have any critical points.

[Smale-1] 1 2 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.

[Conte-2] 1 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.

[Tischler-4] 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.

[Smale2-5] Smale, Steve. "Mathematical Problems for the Next Century" (PDF).

[Crane-6] 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.

[Dubinin-7] 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.

[Ng-8] 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.

Mean value problem

Partial results

See also

Notes

References