Mean value problem

In mathematics, the mean value problem was posed by Stephen Smale in 1981.{{cite journal| last = Smale| first = S.| year = 1981| title = The Fundamental Theorem of Algebra and Complexity Theory| url = https://www.ams.org/journals/bull/1981-04-01/S0273-0979-1981-14858-8/S0273-0979-1981-14858-8.pdf| journal = Bulletin of the American Mathematical Society |series=New Series| volume = 4| issue = 1| pages = 1–36| doi = 10.1090/S0273-0979-1981-14858-8| access-date = 23 October 2017| doi-access = free}} This problem is still open in full generality. The problem asks:

: For a given complex polynomial $f$ of degree $d \ge 2$ {{cite journal| last1 = Conte| first1 = A.| last2 = Fujikawa| first2 = E.| last3 = Lakic| first3 = N.| date = 20 June 2007| title = Smale's mean value conjecture and the coefficients of univalent functions| url = https://www.ams.org/journals/proc/2007-135-10/S0002-9939-07-08861-2/S0002-9939-07-08861-2.pdf| journal = Proceedings of the American Mathematical Society| volume = 135| issue = 10| pages = 3295–3300| doi = 10.1090/S0002-9939-07-08861-2| access-date = 23 October 2017| doi-access = free}}{{efn-ua|text=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.}} and a complex number $z$ , is there a critical point $c$ of $f$ {{nowrap|1=(i.e. $f'(c) = 0$ )}} such that

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

It was proved for $K=4$ . For a polynomial of degree $d$ the constant $K$ has to be at least $\frac{d-1}{d}$ from the example $f(z) = z^{d} - d z$ , therefore no 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, Tischler showed that the conjecture is true for the optimal bound $K = \frac{d-1}{d}$ if $f$ has only real roots, or if all roots of $f$ have the same norm.{{cite journal| last = Tischler| first = D.| year = 1989| title = Critical Points and Values of Complex Polynomials| journal = Journal of Complexity| volume = 5| issue = 4| pages = 438–456| doi = 10.1016/0885-064X(89)90019-8| doi-access = }}{{Cite web|url=http://www6.cityu.edu.hk/ma/doc/people/smales/pap104.pdf|title=Mathematical Problems for the Next Century|last=Smale|first=Steve}}

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

In the same year, Crane showed that $K < 4-\frac{2.263}{\sqrt{d}}$ for $d \ge 8$ .{{cite journal| last = Crane| first = E.| date = 22 August 2007| title = A bound for Smale's mean value conjecture for complex polynomials| url = https://people.maths.bris.ac.uk/~maetc/SMVCbound.pdf| journal = Bulletin of the London Mathematical Society| volume = 39| issue = 5| pages = 781–791| doi = 10.1112/blms/bdm063| s2cid = 59416831| access-date = 23 October 2017}}

Considering the reverse inequality, Dubinin and Sugawa have proven that (under the same conditions as above) there exists a critical point $\zeta$ such that $\left| \frac{f(z) - f(\zeta)}{z - \zeta} \right| \ge \frac$

f'(z)

{n 4^{n}} .{{cite journal| last1 = Dubinin| first1 = V.| last2 = Sugawa| first2 = T.| year = 2009| title = Dual mean value problem for complex polynomials| url = https://projecteuclid.org/euclid.pja/1257430681| journal = Proceedings of the Japan Academy, Series A, Mathematical Sciences| volume = 85| issue = 9| pages = 135–137| doi = 10.3792/pjaa.85.135| access-date = 23 October 2017| arxiv = 0906.4605| bibcode = 2009arXiv0906.4605D| s2cid = 12020364}}

The problem of optimizing this lower bound is known as the dual mean value problem.{{cite journal| last1 = Ng| first1 = T.-W.| last2 = Zhang| first2 = Y.| year = 2016| title = Smale's mean value conjecture for finite Blaschke products| journal = The Journal of Analysis| volume = 24| issue = 2| pages = 331–345| doi = 10.1007/s41478-016-0007-4| arxiv = 1609.00170| bibcode = 2016arXiv160900170N| s2cid = 56272500}}

Notes

References

Category:1981 introductions

Category:Complex numbers

Category:Unsolved problems in mathematics

Category:Conjectures

Mean value problem

Partial results

See also

Notes

References