The Class of Functions Spirallike with Respect to a Boundary Point

The aim of this paper is to present an analytic characterization of the class of functions δ-spirallike with respect to a boundary point. The method of proof is based on Julia's lemma.


Introduction.
In this paper, we study the class * 0 (δ) of δ-spirallike functions with respect to a boundary point.Spirallikeness with respect to a boundary point is a fresh idea being the subject of studies in [1,2].Cited papers developed the method based, on the one hand, on the analytic formula for the class * 0 of functions starlike with respect to a boundary point proposed and proved partially by Robertson [10], and, on the other hand, on some dynamical system built for * 0 (δ).Lyzzaik [8] completing Robertson's proof solved positively his conjecture.Thereby the full analytic description of functions in * 0 was finished.The author [5], by using the Julia lemma, proposed an alternative analytic formula for the class * 0 different than Robertson's characterization.The necessary condition for functions to be in * 0 was shown and, partially, the sufficient condition.In [7], Lyzzaik and the author complete the proof and in this way the class * 0 was equipped with a new analytic characterization.The use of the Julia lemma has the virtue of looking at the inner property of the class * 0 and the other classes defined by the geometric property connected with the boundary point (see, e.g., [6]).In this paper, we apply once again the Julia lemma as a technique to study the class * 0 (δ).Theorem 3.5 demonstrates the basic observation that spirallikeness, as earlier starlikeness with respect to a boundary point, is preserved on each oricycle in the unit disk by every function in * 0 (δ).Theorems 3.6 and 3.8 complete a new analytic characterization of δ-spirallike functions with respect to a boundary point.

Preliminaries. Let
Let Ꮽ denote the set of all analytic functions in D. The subset of Ꮽ of all univalent functions will be denoted by .The set of all ω ∈ Ꮽ such that |ω(z)| < 1 for z ∈ D will be denoted by Ꮾ.
An angular limit of f ∈ Ꮽ at ζ ∈ T will be denoted by f ∠ (ζ).An angular derivative of f ∈ Ꮽ at ζ ∈ T will be denoted by f ∠ (ζ).
For our need, it will be convenient to define the following classes of functions introduced in [5].
Let ᏼ(λ) denote the class of all functions p of the form where ω ∈ Ꮾ(λ).

3.2.
In the proofs of the main theorems of this paper, we will need two lemmas proved in [5].

Lemma 3.3. Every sequence (a n ) of positive numbers with
has a convergent subsequence (a n k ) and (3.3)

3.3.
The theorem below says that every function in * 0 (δ) having a boundary normalization f ∠ (1) = 0 preserves spirallikeness with respect to a boundary point on each oricycle in D. This information will be used later to find an analytic formula for functions in * 0 (δ).
Proof.Assume that f ∈ * 0 (δ) and f ∠ (1) = 0.For each t ≤ 0, define for every t ≤ 0, z ∈ D, and the univalence of f shows that ω t is well defined for each t ≤ 0. Now, fix t < 0 and and and lim n→∞ w n = 0, lim n→∞ z n = 1 by Proposition 3.2.Observe that Let now Hence which means that for each t < 0. In view of Remark 2.2, λ(t) > 0 for every t < 0. Hence, each ω t satisfies the assumptions of the Julia lemma, and since λ(t) ≤ 1 for every t < 0, we derive that Using Theorem 3.5, we characterize functions in * 0 (δ) as follows.
We prove that the last inequality is true for all points on O k for every k > 0. Now, fix k > 0, z ∈ O k and let w = f (z).We parametrize O k as follows: Thus O k is positively oriented.Denote by τ(z) the tangent vector to we have be a parametrization of f (z)L(δ) and let w (0) = lim t→0 − w (t) = e −iδ f (z) be the onesided tangent vector to the logarithmic spiral f (z)L(δ) at f (z).By ϕ(z) we denote the directed angle from the vector iw (0) to τ(z), that is, (3.19)By Theorem 3.5, f (O k ) ∈ ᐆ * 0 (δ) for every k > 0. Hence it is easy to see that where w = f (z) ∈ Γ k .Indeed, let w 0 ∈ wL(δ) be arbitrary.Thus w 0 = wu 0 for some u 0 ∈ L(δ).Since w ∈ Γ k , there exists a sequence (w n ) of points in f (O k ) convergent to w.The inclusion w n L(δ) ⊂ f (O k ) yields that w n u 0 is a point of f (O k ) for every n ∈ N. At the end, the convergence of the sequence (w n u 0 ) of points of f (O k ) to w 0 implies that w 0 ∈ f (O k ).Since w 0 was arbitrary, our claim is proved.
Let l be a line going through f (z) with w (0) as the directional vector.Then l divides the plane into two closed half-planes H 1 and H 2 .One of them, say H 1 , contains the origin and the spiral f (z)L(δ).We assume first that δ ∈ (−π/2, 0).This means that the spiral L(δ) has the shape such that it attains 1 from the lower half-plane.Moreover, f (z)L(δ) parametrized as above turns round the origin in the counterclockwise direction.Hence, iw (0) lies in H 1 .By Theorem 3.5, f (O k ) ∈ ᐆ * 0 (δ).Hence, and from (3.20), it follows that either Γ k is tangent both to f (z)L(δ) (one-sided) and to l at f (z), and then τ(z) lies in l so in H 1 , or by [9, Proposition 2.13, page 28], there is a crosscut C ⊂ l of f (O k ) with one endpoint at f (z).Thus, by [9, Proposition 2.12, page 27], f (O k ) has exactly two components, one of them, say G, lies in H 2 .Clearly, ∂G = C ∪ Γ , where Γ ⊂ Γ k ends at f (z).Hence Γ is a subset of H 2 and, since it is part of a positively oriented closed analytic curve Γ k , we deduce finally that the tangent vector τ(z) to Γ k at f (z) lies in H 1 .In a similar way, we can prove that both vectors iw (0) and τ(z) lie together in H 2 as δ ∈ (0,π/2).This, (3.19), and the fact that iw (0) is orthogonal to l yield As k > 0 and z ∈ O k was arbitrary, this is true in D. Suppose now that equality holds in (3.21) for some z 0 ∈ D. By the maximum principle for harmonic functions, it holds in the whole disk D, which implies that there exists y ∈ R \{0} so that But the solution Then ω(D) ⊂ D. We now prove that ω ∈ Ꮾ(λ) for some λ ∈ (0, 1].Recalling the Visser-Ostrowski quotient, we can write Since, for every r ∈ (0, 1), so we can find a subsequence (r n k ) of (r n ) such that Hence ω satisfies the assumptions of the Julia lemma with λ = λ 1 .Since then (2.5) holds, by using Lemma 2.3 and Remark 2.4, we see that ω ∈ Ꮾ(Λ), where Λ ≤ λ 1 ≤ 1 is given by (2.7).This ends the proof of the theorem.
for every Stolz angle ∆.

Proof.
Since for every Stolz angle ∆, the assertion follows at once with λ = Λ.
Proof.First we show that f is univalent in D. It is immediate from (3.13) that f is locally univalent in D. Let ω ∈ Ꮾ(λ), where λ ∈ (0, cos δ], and let g be the solution of the differential equation with the boundary condition g ∠ (1) = 0.As was proved in [7, Theorem 3], g belongs to the class * 0 , so it is univalent, and g(D) being a simply connected domain lies in a wedge of angle 2λπ .Hence there exists a single-valued analytic branch of log g in D, and is well defined.But, in view of (3.13) and (3.34), we have (3.37) Since λ ∈ (0, cos δ], from the above, the univalence of f in D follows.Now, we prove that f (D) ∈ ᐆ * 0 (δ).This is clear, looking at the relation (3.37) between classes * 0 and * 0 (δ), which yields the geometric relation between starlikeness and spirallikeness of domains in the plane.To be self-contained, we prove it without using geometric properties of functions in * 0 .We assume that δ = 0, since this case reduces to [7,Theorem 3].
Remark 3.9.In [1], the authors found necessary and sufficient conditions for functions to be in * 0 (δ) (Theorem 2.1).The analytic formula (2.1) in [1] generalizes the Robertson inequality for starlike functions with respect to a boundary point.In fact, the authors of [1] proved that each spirallike function with respect to a boundary point is a complex power of a corresponding function which is starlike with respect to a boundary point.Formula (3.13) presents an alternative analytic description of the class * 0 (δ).In case δ = 0 (µ = 2π in [1, equation (2.1)]), these two analytic formulas for * 0 (0) characterizing starlike functions with respect to a boundary point are equivalent.Looking at [1, Theorem 2.1(III) and Theorems 3.2 and 3.3], we can expect that formulas (2.1) in [1] and (3.13) of the present paper are equivalent, which, in fact, means that in Theorem 3.6 the assumptions λ ∈ (0, 1] should be replaced by λ ∈ (0, cos δ].This is an open problem.
has the shape such that it attains 1 from the lower half-plane.Moreover, w 1 L(δ) parametrized as above turns round the origin in the counterclockwise direction.Hence, iw (0) lies in H 1 .Observe that either Γ k is tangent as well to w 1 L(δ) (one-sided) as to l at w 1 and then τ(z 1 ) lies in l, or, by [9, Proposition 2.13, page 28], there is a crosscut C ⊂ l of f (O k ) with one endpoint at w 1 .Thus, by [9, Proposition 2.12, page 27], f (O k ) has exactly two components, one of them, say G, lies in H 2 .Moreover, ∂G = C ∪Γ , where Γ ⊂ Γ k ends at w 1 .Hence Γ is a subset of H 2 and, since it is part of a positively oriented closed analytic curve Γ k , we deduce finally that the tangent vector τ(z 1 ) to Γ k at f (z 1 ) lies in H 2 .Since iw (0) is orthogonal to l and lies in H 1 , we deduce that Therefore the tangent line l to w 0 L(δ) at w 1 has the directional vector v (t 1 ) = w (0) and is the boundary of two closed half-planes denoted by H 1 and H 2 .One of them, say H 1 , contains the origin.Let δ ∈ (−π/2, 0).As we remarked in the proof of Theorem 3.6, the spiral L(δ)