Partial sums of functions of bounded turning

In [6] it was shown that if f ∈ is starlike of order α, α= 0.294, . . . , so is the Libera integral operator F . We also know that (see, e.g., [1]) there are functions which are univalent or spiral-like in so that their Libera integral operators are not univalent or spiral-like in . Li and Owa [5] proved that if f ∈ is univalent in , then Fn(z) is starlike in |z| < 3/8. The number 3/8 is sharp. In this note we make use of a result of Gasper [2] to provide a simple proof for the following theorem. Main theorem. If 1/4≤α< 1 and f ∈ (α), then Fn ∈ ((4α−1)/3).