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|
|Pages (from-to)||781 - 791|
|Number of pages||11|
|Journal||Bulletin of the London Mathematical Society|
|Publication status||Published - Oct 2007|