We improve an estimate of the constant in Smale's mean value conjecture, by using the Bieberbach theorem for coefficients of univalent functions and an estimate of the hyperbolic density of a certain simply connected domain.
Ege Fujikawa. Toshiyuki Sugawa. "Geometric function theory and Smale's mean value conjecture." Proc. Japan Acad. Ser. A Math. Sci. 82 (7) 97 - 100, March 2006. https://doi.org/10.3792/pjaa.82.97