Abstract
For a prestarlike function
f
of nonnegative order
α
,
0
≤
α
<
1
, and a close-to-convex function
zg
of order
α
, the convolution
g
∗
f
′
is shown to be zero-free in the open unit disk. The result can be applied to a wide spectrum of interesting approximants, including those involving the Cesàro means and Jacobi polynomials. If
zg
is also prestarlike, then the range of
g
∗
f
′
is shown to be contained in a sector with opening angle strictly less than 2
π
.
MSC:
30C45, 33C05, 40G05, 41A10.