On the radius of convexity and boundary distortion of Schlicht functions

Let $w = f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}}$ be regular and univalent for $\vert z\vert < 1$ and map $\vert z\vert < 1$ onto a region which is starlike with respect to $w = 0$. If ${r_0}$ denotes the radius of convexity of $w = f(z)$, $d_0 = \min \vert f(z)\vert$ for $\vert z\vert = {r_0}$, and ${d^ \ast } = \inf \vert\beta \vert$ for $f(z) \ne \beta$, then it has been conjectured that ${d_0}/{d^ \ast } \geqq 2/3$. It is shown here that ${d_0}/{d^ \ast } \geqq 0.343 \ldots$, which improves the old estimate ${d_0}/{d^ \ast } \geqq 0.268 \ldots$. In addition, sharp estimates for ${r_0}$ are given which depend on the value of $\vert{a_2}\vert$.