Angle distortion theorems for starlike and spirallike functions with respect to a boundary point

It is clear that 0∈ h(Δ). Moreover, (i) if 0 ∈ h(Δ), then h is called spirallike (resp., starlike) with respect to an interior point; (ii) if 0 ∈ h(Δ), then h is called spirallike (resp., starlike) with respect to a boundary point. In this case, there is a boundary point (say, z = 1) such that h(1) := ∠ limz→1h(z)= 0 (see, e.g., [1, 6]); by symbol∠ lim, we denote the angular (nontangential) limit of a function at a boundary point of Δ. The class of spirallike (starlike) functions with respect to a boundary point normalized by the conditions h(1)= 0 and h(0)= 1 will be denoted by Spiral [1] (resp., Star [1]). It was proved in [1, 7] that for any function h ∈ Spiral [1], the limit of the so-called Visser-Ostrowski quotient