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


Introduction
Let Δ be the open unit disk in the complex plane C. By Hol(Δ,C) we denote the set of all holomorphic functions in Δ.
It turns out that the number μ can be used for some geometric characteristic of the image h(Δ).Namely, it was shown in [7] that if μ is real, then the smallest wedge with vertex at the origin which contains h(Δ) is exactly of angle μπ.
In this paper, we establish an angle distortion theorem for a class of unbounded starlike and spirallike functions which gives us both covering and growth estimates.
, then the wedge where is contained in h(Δ), and there is no wedge Conversely, if h ∈ Star μ [1] and the image h(Δ) contains some open wedge, then there are ν (−μ ≤ ν < 0) and an automorphism Ψ of the open unit disk Δ such that function

Also one can write
where the probability measures dσ and δ(−1) are mutually singular.Using this decomposition, we rewrite (2.8) as follows: This formula coincides with (2.5) for κ = −bμ.
To prove the converse assertion, suppose that the open wedge lies in h(Δ).Then the curve := h −1 ({w : argw = α}) lies in Δ and joins the point 1 with another boundary point η.Denote by Ψ an automorphism of Δ such that Ψ(1) = 1 and Consequently, h 1 is a starlike function with respect to a boundary point, and where

.29)
A simple calculation shows that h 1 ∈ Star μ [1] and Since W 1 cannot be extended to a larger wedge lying in h 1 (Δ), for each ε > 0 there are boundary points of the image h 1 (Δ) belonging to (2.30) Let {φ + n } be a decreasing sequence such that φ + n → argη, the values h 1 (e iφ + n ) exist, and Similarly, let {φ − n } be an increasing sequence such that φ − n → argη, the values h 1 (e iφ − n ) exist, and where the measure dσ is singular relative to δ(−1) and −μ ≤ κ ≤ 0. Consider the expression arg h 1 (e iφ + n ) − argh 1 (ie φ − n ) which tends to 2β: (2.32) The first summand tends to zero while the second one tends to −κπ.The third summand also tends to zero because the integrand is a bounded function which tends to zero for each ζ ∈ ∂Δ, ζ = 1.Hence by Lebesgue's bounded convergence theorem, the integral in (2.32) goes to 0.

Functions convex in one direction
The class of functions we consider here has been studied by several mathematicians (see, e.g., Ciozda [3,4], Hengartner and Schober [11], Lecko [12]) as a subclass of functions defined by Robertson in [14].

Definition 3.1. Say that a univalent function h ∈ Hol(Δ,C) normalized by
is convex in the positive direction of the real axis if for each z ∈ Δ and t > 0, The class of those functions is denoted by Σ [1].
Here we find the maximal width size of the image for functions of this class using an angular limit characteristic of functions under consideration.In other words, given such a function, we find the minimal horizontal strip which contains its image.The following question is also natural but more complicated: characterize those functions convex in the positive direction of the real axis whose images contain a whole (two-sided) strip and find the size (width) of this strip.We solve this problem for functions having maximal horizontal strips of finite size.The problem is still open for the general case.
To proceed, we need the following lemma.
Lemma 3.2 (cf.[3,4,11]).Let h be a univalent function normalized by (3.1).Then h ∈ Σ [1] if and only if . By Definition 3.1, for each t ≥ 0, the holomorphic function F t defined by maps the unit disk into itself.It is easy to verify that the family = F t t≥0 forms a continuous semigroup of holomorphic self-mappings of the unit disk.
Differentiating this semigroup at t = 0 + , we get where f is the so-called infinitesimal generator f of (see, e.g., [16]).By (3.2) the point τ = 1 is the Denjoy-Wolff point of (see, e.g., [2,16]).Therefore, its generator can be represented by the Berkson-Porta formula (see [2]): If Re p(z) > 0, z ∈ Δ, then by a result of Berkson and Porta (see [2]), the function f defined by (3.6) is the generator of a semigroup = F t t≥0 of holomorphic self-mappings of the unit disk.This semigroup can be defined by the Cauchy problem: Integrating the latter expression on the interval [0, t], we get M. Elin and D. Shoikhet 9 Since has a Denjoy-Wolff point at 1, it follows that This completes our proof.
is either a positive real number or infinity.
Proof.By Lemma 3.2, the function is either of positive real part or an imaginary constant.In the latter case, p(z) = ib, b ∈ R, and the assertion is evident.Otherwise, one can write It was proved in [8] (see also [17]) that for any function p with positive real part, the angular limit ∠ lim z→1 (1 − z)p(z) exists and is a nonnegative real number.This proves our assertion.
is contained in h(Δ), and there is no strip Proof.(i) Suppose that h ∈ Σ μ [1], μ ∈ (0,∞), that is, the limit (3.11) is finite.By Lemma 3.2, the function p(z) = (1 − z) 2 h (z) is of nonnegative real part.Hence, there exists a self-mapping ω(z) such that Since the right-hand side of this equality is bounded on each nontangential approach region at the point z = 1, one concludes that ∠ lim z→1 ω(z) = 1.Now we calculate Consider the function  we see that g ∈ Star λ [1] with λ = 1.By Theorem 2.2, the smallest wedge which contains the image g(Δ) is defined by (2.2): where

.26)
Therefore the smallest horizontal strip which contains the image of function h(z) = −μ log g(z) is defined by (3.17).Suppose now that h ∈ Σ [1] and its image is contained in a horizontal strip of finite width πμ (0 < μ < ∞).In this case, the function g defined by (3.22) is univalent and starlike with respect to a boundary point.Moreover, since the minimal strip which contains the image h(Δ) is of width πμ, the minimal wedge containing g(Δ) is of angle π.Once again by Theorem 2.2, g ∈ Star λ [1] with λ = 1; hence it satisfies the Visser-Ostrowski condition is contained in g(Δ), and there is no wedge W = W * with W * ⊂ W ⊂ g(Δ).Since h(z) = −μ log g(z), we have that the limit a * = lim x→−1 + Im h(x) = −μθ * exists.In addition, the strip S * defined by (3.18) is contained in h(Δ), and there is no strip S = S * with S * ⊂ S ⊂ h(Δ).
Remark 3.8.Unfortunately, the problem of finding a maximal strip which is contained in the image h(Δ) is still open when h ∈ Σ ∞ [1].This problem is a key to solve the following important problem in the theory of semigroups of parabolic type.
If = {F t } t 0 has a Denjoy-Wolff point τ ∈ ∂Δ of parabolic type, that is, F t (τ) = 1 for all t ≥ 0, then there is a solution of Abel's functional equation h F t (z) = h(z) + t (3.32) (see, e.g., [10]) which is of the class Σ ∞ [1].If = {F t } t 0 has a repelling boundary fixed point, then this problem is equivalent to the existence of an open strip which is contained in h(Δ).