Computational Methods and Function Theory 6 (2006), No. 1, 145--163
Copyright Heldermann Verlag 2006
Edward Crane
edward.crane@gmail.com , Mathematics Department, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom.
Let p be a non-linear complex polynomial in one variable. Smale's mean value conjecture is a precise estimate of the derivative p'(z) in terms of the gradients of chords between z and a stationary point on the graph of p. The problem is to determine the correct constant in the estimate, but despite the apparent simplicity of the problem only a small amount of progress has been made since Stephen Smale first posed it in 1981. In this paper we establish the existence of extremal polynomials for Smale's mean value conjecture, and establish a geometric property of the extremals.
Keywords: Complex polynomials, critical points, minimax problems.
MSC 2000: Primary 26C10; Secondary 30C15, 49J35.
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