A note on a problem of Robinson

Abstract:

Let $\mathcal {S}$ be the usual class of univalent analytic functions on $\left | z \right | < 1$ normalized by $f(0) = 0$ and $f’(0) = 1$. Let $\mathfrak {L}$ be the linear operator on $\mathcal {S}$ given by $\mathfrak {L}f = \tfrac {1}{2}(zf)’$ and let ${r_{{\mathcal {S}_t}}}$ be the minimum radius of starlikeness of $\mathfrak {L}f$ for $f$ in $\mathcal {S}$. In 1947 R. M. Robinson initiated the study of properties of $\mathfrak {L}$ acting on $\mathcal {S}$ when he showed that ${r_{{\mathcal {S}_t}}} > .38$. Later, in 1975, R. W. Barnard gave an example which showed ${r_{{\mathcal {S}_t}}} < .445$. It is shown here, using a distortion theorem and Jenkin’s region of variability for $zf’(z)/f(z)$, $f$ in $\mathcal {S}$, that ${r_{{\mathcal {S}_t}}} > .435$. Also, a simple example, a close-to-convex half-line mapping, is given which again shows ${r_{{\mathcal {S}_t}}} < .445$.