Dual Smale’s mean value conjecture
HTML articles powered by AMS MathViewer
- by Aimo Hinkkanen, Ilgiz R. Kayumov and Diana M. Khammatova PDF
- Proc. Amer. Math. Soc. 147 (2019), 5227-5237 Request permission
Abstract:
We prove the dual Smale’s mean value conjecture for polynomials of degree six: if $f$ is a polynomial of degree six with $f(0)=0$ and $f’(0)=1$, then there is a point $\zeta$ such that $f’(\zeta )=0$ and $| f(\zeta )/\zeta | \geq 1/6$.References
- A. F. Beardon, D. Minda, and T. W. Ng, Smale’s mean value conjecture and the hyperbolic metric, Math. Ann. 322 (2002), no. 4, 623–632. MR 1905109, DOI 10.1007/s002080000184
- V. N. Dubinin, Methods of geometric function theory in classical and modern problems for polynomials, Uspekhi Mat. Nauk 67 (2012), no. 4(406), 3–88 (Russian, with Russian summary); English transl., Russian Math. Surveys 67 (2012), no. 4, 599–684. MR 3013845, DOI 10.1070/RM2012v067n04ABEH004803
- Vladimir Dubinin and Toshiyuki Sugawa, Dual mean value problem for complex polynomials, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 9, 135–137. MR 2573962
- Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1–36. MR 590817, DOI 10.1090/S0273-0979-1981-14858-8
- David Tischler, Critical points and values of complex polynomials, J. Complexity 5 (1989), no. 4, 438–456. MR 1028906, DOI 10.1016/0885-064X(89)90019-8
- Jeremy T. Tyson, Counterexamples to Tischler’s strong form of Smale’s mean value conjecture, Bull. London Math. Soc. 37 (2005), no. 1, 95–106. MR 2105824, DOI 10.1112/S0024609304003613
Additional Information
- Aimo Hinkkanen
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801-2975
- MR Author ID: 86135
- Email: aimo@math.uiuc.edu
- Ilgiz R. Kayumov
- Affiliation: Kazan Federal University, Kremlevskaya 18, 420 008 Kazan, Russia
- Email: ikayumov@kpfu.ru
- Diana M. Khammatova
- Affiliation: Kazan Federal University, Kremlevskaya 18, 420 008 Kazan, Russia
- Email: dianalynx@rambler.ru
- Received by editor(s): May 28, 2017
- Received by editor(s) in revised form: December 21, 2018, and March 6, 2019
- Published electronically: July 8, 2019
- Additional Notes: The research of the second and third authors was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.13556.2019/13.1.
- Communicated by: Jeremy Tyson
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5227-5237
- MSC (2010): Primary 30C10
- DOI: https://doi.org/10.1090/proc/14639
- MathSciNet review: 4021082