Harmonic Archimedean and hyperbolic spirallikeness

We define a harmonic functions called Archimedean spirallike and hyperbolic spirallike functions. We investigate their geometric and analytic properties. Some examples are provided.


Introduction
Let D r = {z ∈ C : |z| < r } be the open disc of the radius r of the complex plane, T r = {z ∈ C : |z| = r } and let D 1 = D be the unit disk. Also, we denote by A the class of analytic functions on D with standard normalization f (0) = f (0) − 1 = 0.
A harmonic mapping f of the simply connected region is a complex-valued function of the form f = h + g, (1.1) S.Kanas by coefficients of the power series expansions [3] h(z) = a 0 + ∞ n=1 a n z n , g(z) = b 0 + ∞ n=1 b n z n (z ∈ D), (1.2) where a n ∈ C, n = 0, 1, 2, ... and b n ∈ C, n = 1, 2, 3, .... By H 0 a subclass of H with the normalization h(0) = g(0) = 0, h (0) = 1. Following Clunie and Sheil-Small notation [3], we denote by S H the subclass of H 0 , consisting of all sense-preserving univalent harmonic mappings of D. Several fundamental information about harmonic mappings in the plane can also be found in [4].

Differential operators
For f ∈ C 1 (D), let the differential operators D and D be defined as follows and where ∂ f /∂z and ∂ f /∂z are the formal derivatives of the function f Moreover, we define n-th order differential operator by the recurrence relation We note that in the case when f is an analytic function (i.e. g(z) = 0), then both D and D reduce to the Alexander differential operator z f . Now, we present several properties of the differential operators D f and D f . Most of them follow from the usual rules of differential calculus therefore the proofs will be omitted. Proposition 2.1 Let ϕ, ψ ∈ C 1 (D) and let the linear differential operators D and D be defined by (2.1) and (2.2). Then: Dψ.

Proposition 2.2
Let f ∈ C 1 (D) and let D and D be defined by (2.1) and (2.2). Then

Remark 2.4
If G ∈ H, then DG(zz) = 0 and DG(arg z) = 0. Therefore the constant functions for the operators D and D are the functions of the form G(|z| 2 ) and G(arg z), respectively.

Starlikeness and spirallikenes of analytic functions
A domain D ⊂ C is said to be starlike w.r.t. origin if each point w ∈ D may be connected with origin by a segment that lies entirely in D. Geometrically, this means that the linear segment joining the origin to every other point w lies entirely in D.
An analytic function f that maps the unit disk D onto starlike domain is called starlike function [9]. Every starlike function in A is necessarily univalent. An analytic necessary and sufficient condition for starlikeness of univalent functions is: Modifying the starlikeness condition by inserting a factor e iγ (|γ | < π/2) we obtain that is the condition of γ -spirallikeness of analytic functions f in D. The notion of γ -spirallikeness of f (D) geometrically means that the arc of the logarithmic spiral (σ t ) = te iγ (t ∈ [0, ∞)) joining the origin to every other point w lies entirely in f (D). It was shown by Spaček [8] that spirallike functions are univalent. Gamma spirallike functions gained recognition of many researchers, their generalizations were introduced and many properties were studied (see, for example [2,6,10]).
In 1981 Al-Amiri and Mocanu [1] proved a sufficient condition for a function f ∈ C 1 (D) to be univalent and to map D onto a spirallike domain.
Suppose that a function f ∈ C 1 (D) that vanishes only at the origin, and let γ be a given real number such that |γ | < π/2. If J f > 0 on D, and It is noteworthy that (3.2) reduces then to (3.1) in the case of f ∈ A. The properties of harmonic starlike and spirallike functions were considered in [7].

Harmonic Archimedean and hyperbolic spirallikeness
Al-Amiri and Mocanu in their paper [1] stated that the same method of proof for γ -spirallikeness can be used to show a sufficient conditions for Archimedean and hyperbolic starlikeness.
is satisfied.
Proof The proof will be a modification and supplement to that from [1], which concerned the γ -spirallikeness conditions of f ∈ C 1 (D), and contained only necessary condition for γ -spirallikeness. Assume first that (4.1) is satisfied. For 0 < r < 1 we denote C r = f (T r ). We note that 0 / ∈ C r for 0 < r < 1. We now prove that the function f is univalent in D. To do this we will show that (C r ) contains only non-intersecting Jordan curves. Let (σ φ ) be the family of spirals such that σ φ has the parametric representation σ φ : w = w φ (t), t ∈ R, and w φ = te i(t+φ) . It is clear that through each point z ∈ C \ {0} passes only one spiral of the family (σ φ ). Hence, for z = re iθ (0 < r < 1, 0 ≤ θ < 2π ), the equation f (z) = w φ (t) determines a unique φ = φ(r , θ) ∈ [0, 2π). We first prove that C r is a Jordan curve for each 0 < r < 1. It can be achieved by showing that and that the total variation of φ(r , θ) on a segment [0, 2π) is equal 2π . From the representation of w φ we get (4.3) and from this Differentiating with respect to θ and using (2.4) and (2.5) we obtain from (4.4) Hence, by (4.1) the condition (4.2) is satisfied.
Furthermore, condition f (z) = 0 for z ∈ D \ {0} implies that the curves C r , r ∈ (0, 1), are homotopic in the domain C \ {0}. Thus they have the same index with respect to the origin, i.e., ind 0 C r = const for all r ∈ (0, 1). By condition J f > 0 the function f is univalent and preserves the orientation in a neighborhood of the origin. This implies the existence of r 0 ∈ (0, 1) such that ind 0 C r = 1 for r < r 0 . Hence the total variation of the argument along C r is 2π , that is, ∈ (0, 1)). which gives that for each r ∈ (0, 1), C r is a simple Archimedean spirallike.

Remark 4.9
We note that the condition (4.12) can be rewritten as

Examples
The introduced function classes are not empty, even though it is not easy to determine the appropriate examples. Below we present some examples of the functions of the considered classes.
Example 5. 1 We note that the harmonic Koebe function does not satisfy the condition (4.1), that is the harmonic Koebe function is not Archimedean spirallike. Also, harmonic Koebe function is not hyperbolic spirallike. Indeed, for k H (z) = h(z) + g(z), where and for Archimedean case we obtain (using Wolfram Mathematica, ver. 8.0) and for z 0 = 1 2 − i 2 in the hyperbolic case we have which means, that generalized spirallikeness is possible, for some G. Re which means that f is not hyperbolic spirallike.

Example 5.3
Consider now the function mapping the unit disk to a domain similar in shape to the mapping from the previous example ( Fig. 2) f (z) = z (1 − z/5)(1 − z/5) .