A note on the $2/3$ conjecture for starlike functions

Abstract:

Let $w = f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}$ be regular and univalent for $|z| < 1$ and map $|z| < 1$ onto a region which is starlike with respect to $w = 0$. If ${r_0}$ denotes the radius of convexity of $w = f(z),{d_0} = \min |f(z)|$ for $|z| = {r_0}$, and ${d^ \ast } = \inf |\beta |$ for $f(z) \ne \beta$, then it has been conjectured that ${d_0}/{d^ \ast } \geqq 2/3$. It is shown here that ${d_0}/{d^\ast } \geqq 0.380 \cdots$ which improves the old estimate ${d_0}/{d^\ast } \geqq 0.343 \cdots$. In addition an upper bound for ${d^ \ast }$ which depends on $|{a_2}|$ is given.