## Abstract

Smale's mean value conjecture asserts that min_{θ} |P(θ)/θ| ≤ K|P′(0)| for every polynomial P of degree d satisfying P(0) = 0, where K = (d-1)/d and the minimum is taken over all critical points 6 of P. A stronger conjecture due to Tischler asserts that min_{θ}|1/2 - P(θ)/θ·P′(0)| ≤ K _{1} with K_{1} = 1/2 - 1/d. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum P_{0}(z) = z ^{d} - dz, and (ii) for all polynomials of degree d ≤ 4. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum P_{1}(z) = (z - 1)^{d} -(-1)^{d}, but counterexamples to the conjecture are given in each degree d ≥ 5. In addition, estimates for certain weighted L^{1}- and L ^{2}-averages of the quantities 1/2 - P(θ)/θ · P′(0) are established, which lead to the best currently known value for K_{1} in the case d = 5.

Original language | English (US) |
---|---|

Pages (from-to) | 95-106 |

Number of pages | 12 |

Journal | Bulletin of the London Mathematical Society |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2005 |

## ASJC Scopus subject areas

- Mathematics(all)