## Abstract

Smale's mean value conjecture is an inequality that relates the locations of critcal points and critical avalues of a polynomial p to the value and derivative of p at some given non-critical point. Using known estimatates for the logarithmic capacity of a connected set in the plane containing three given points, we give a new bound for the constant in Smale's inequality in terms of the degree d of p. The bound improves previous results when d > 8.

Translated title of the contribution | A bound for Smale's mean value conjecture for complex polynomials |
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Original language | English |

Pages (from-to) | 781 - 791 |

Number of pages | 11 |

Journal | Bulletin of the London Mathematical Society |

Volume | 39 (5) |

DOIs | |

Publication status | Published - Oct 2007 |