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A bound for Smale's mean value conjecture for complex polynomials
Author(s) -
Crane Edward
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm063
Subject(s) - mathematics , conjecture , logarithm , upper and lower bounds , combinatorics , value (mathematics) , constant (computer programming) , polynomial , critical point (mathematics) , logarithmic derivative , plane (geometry) , derivative (finance) , complex plane , degree (music) , set (abstract data type) , mathematical analysis , statistics , geometry , physics , computer science , acoustics , financial economics , economics , programming language
Smale's mean value conjecture is an inequality that relates the locations of critical points and critical values of a polynomial p to the value and derivative of p at some given non‐critical point. Using known estimates 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.