A counterexample to the two-thirds conjecture

Abstract:

Let $w = f(z) = z + {a_2}{z^2} + \cdots$ be regular and univalent for $|z| < 1$, and $\operatorname {map} |z| < 1$ onto a region which is starlike with respect to $w = 0$. If ${r_0}$ denotes the radius of convexity of $f(z),{d^\ast } = \min |f(z)|$ for $|z| = {r_0}$ and $d = \inf |\beta |$ for which $f(z) \ne \beta$, then it has been conjectured by A. Schild in 1953 that ${d^\ast }/d \geqq \tfrac {2}{3}$. It is shown here that this conjecture is false by giving two counter-examples.