Digital Object Identifier (DOI) 10.1007/s002080000184 Math. Ann. 322, 623–632 (2002) Mathematische Annalen Smale’s MeanValue conjecture and the hyperbolic metric A.F. Beardon · D. Minda · T.W. Ng Received: 9 May 2000 / Published online: 28 February 2002 – c  Springer-Verlag 2002 Mathematics Subject Classification (1991): 30C15, 30F45, 30C20 1. Smale’s MeanValue conjecture Let P be any polynomial; then z is a critical point of P if and only if P ′ (z) = 0, and w is a critical value of P if and only if w = P(z) for some critical point z of P . A nonconstant linear polynomial has no critical points, so throughout the paper we shall be assuming that P has degree d , where d ≥ 2. We begin with a result and conjecture of Smale. Theorem 1.1 [5]. Let P be a non-linear polynomial with critical points z j . If z is not a critical point of P then min j     P(z) - P(z j ) z - z j     ≤ 4|P ′ (z)|. (1.1) Smale proved this in 1981 ([5],p.33), and then asked whether one can replace the factor 4 in the upper bound in (1.1) by 1, or even possibly by (d - 1)/d . He repeated this problem in [6] (p.289, although not in his list of major problems). The number (d - 1)/d would, if true, be the best possible bound here as it is A.F. Beardon Department of Pure Mathematics and Math. Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: afb@dpmms.cam.ac.uk) D. Minda Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA (e-mail: David.Minda@math.uc.edu) T.W. Ng Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong (e-mail: ntw@maths.hku.hk)