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