For β ≧ 0, denote by K(β)
the class of normalized functions f, regular and locally schlicht in the unit disc,
which satisfy the condition that for each r < 1, the tangent to the curve
C(r) = {f(rei𝜃) : 0 ≦ 𝜃 < 2π} never turns back on itself as much as βπ radians. K(β)
is called the class of close-to-convex functions of order β. The purpose of this
paper is to investigate the asymptotic behavior of the integral means and
Taylor coefficients of f ∈ K(β). It is shown that the function Fβ, given by
Fβ(z) = (1∕(2(β + 1))){((1 + z)∕(1 −z))β+1− 1}, is in some sense extremal for each
of these problems. In addition, the class B(α) of Bazilevic functions of type α > 0 is
related to the class K(1∕α). This leads to a simple geometric interpretation of the
class B(α) as well as a geometric proof that B(α) contains only schlicht
functions.
