Integral transforms of functions with the derivative in a halfplane

Let A be the class of normalized analytic functions in the unit disk Δ and define the class Pβ = {f ∈ A| ∃α ∈ ℝ| Re{eia(f′(z) - β)} > 0, z ∈ Δ}. For a function f ∈ A the Alexander transform F0 is given by F0(z) - ∫10 f(tz)/tdt. Our main object is to establish a sharp relation between β and γ such that f ∈ Pβ implies that F0 is starlike of order γ, 0 ≤ γ ≤ 1/2. A corresponding result for the Libera transform F1(z) = 2 ∫10 f (tz)dt is also given.