Continua
of minimal capacity and
Smale's
mean value conjecture
Dr. Edward Crane
Abstract
Smale's mean
value inequality is a constraint on the location of critical values and
critical points of a complex polynomial P
with degree d ³ 2.
It gives a lower
bound for P'(z) in terms of the gradients of chords on the graph of P from the point (z, P(z)) to the stationary points (z, P(z)), where P'(z) = 0.
The mean value
conjecture concerns the multiplicative constant in the inequality. It
says that the example z = 0, P(z) = z^{d}  z should be extremal
among polynomials of degree d.
We will describe a
recent improvement in the constant, which relies on results of Jenkins and Kuzmina giving a lower bound for the logarithmic capacity
of a plane continuum containing three given points.
Date: 
June 4, 2008 (Wednesday) 
Time: 
2:30 – 3:30pm 
Place: 
Room 517, Meng Wah Complex, HKU 

All are
welcome 
