Let S \mathcal {S} be the usual class of univalent analytic functions on | z | > 1 \left | z \right | > 1 normalized by f ( 0 ) = 0 f(0) = 0 and f ′ ( 0 ) = 1 f’(0) = 1 . Let L \mathfrak {L} be the linear operator on S \mathcal {S} given by L f = 1 2 ( z f ) ′ \mathfrak {L}f = \tfrac {1}{2}(zf)’ and let r S t {r_{{\mathcal {S}_t}}} be the minimum radius of starlikeness of L f \mathfrak {L}f for f f in S \mathcal {S} . In 1947 R. M. Robinson initiated the study of properties of L \mathfrak {L} acting on S \mathcal {S} when he showed that r S t > .38 {r_{{\mathcal {S}_t}}} > .38 . Later, in 1975, R. W. Barnard gave an example which showed r S t > .445 {r_{{\mathcal {S}_t}}} > .445 . It is shown here, using a distortion theorem and Jenkin’s region of variability for z f ′ ( z ) / f ( z ) zf’(z)/f(z) , f f in S \mathcal {S} , that r S t > .435 {r_{{\mathcal {S}_t}}} > .435 . Also, a simple example, a close-to-convex half-line mapping, is given which again shows r S t > .445 {r_{{\mathcal {S}_t}}} > .445 .