On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc ∆. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998–2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6], among others. This work continues the investigations described in [7]. Our results relate primarily to the order of starlikeness and convexity of functions of the aforementioned classes.

In this paper we will show results related to the order of starlikeness and convexity of functions which belong to the aforementioned classes.
Recall the definition of complex harmonic functions starlike (convex) and starlike (convex) of the order β, β ∈ [0, 1).Definition 3. A univalent and sense-preserving complex harmonic function f of the form (1.1) is called starlike with respect to the origin (starlike) in Δ, if f (Δ) is a domain starlike with respect to the origin.Definition 4. A univalent and sense-preserving complex harmonic function f of the form (1.1) Remark 1.It is known that in order to prove starlikeness of the image of the disc Δ by a univalent sense-preserving mapping f it is sufficient to prove starlikeness of f (Δ r ) for every r ∈ (0, 1), i.e. to show that for any r ∈ (0, 1) By analogy, in order to prove convexity of the image of the disc Δ by a univalent sense-preserving mapping f , it is sufficient to prove convexity of f (Δ r ) for every r ∈ (0, 1), i.e. to show that for any r ∈ (0, 1) we have ).A univalent and sense-preserving complex harmonic function f of the form (1.1) is called starlike of the order β with respect to the origin in Δ if for any r ∈ (0, 1), we have ).A univalent and sense-preserving complex harmonic function f of the form (1.1) is called convex of the order β in Δ if for any r ∈ (0, 1), we have Remark 2. Definitions 5 and 6 are analogues of the appropriate classical definitions for the normalized holomorphic functions in Δ.In [9], among other things, the definition of a starlike (convex) function f of the form and for any sufficiently small > 0 there exists a point The second part of Robertson's definition is overlooked by his followers.
In this paper we use generalization of the classical definition of the functions starlike (convex) of the order β, for complex harmonic functions.
It is worth remembering that the property of starlikeness (convexity) is hereditary for functions holomorphic in Δ, but for complex harmonic functions it need not be so [4].

Let us denote
In [7] the following theorems are proved:

Then HS(α, p) is the class of univalent and sense-preserving functions in Δ.
A 1 is the largest set, in which every function Remark 3. The sets A 1 and A 2 are the largest subsets of A, in which every function f ∈ HS 0 (α, p) is starlike, convex in Δ, respectively.

Main results.
In view of Theorem 2 it seems natural to ask a question about the order of starlikeness of functions of the class HS 0 (α, p), (α, p) ∈ A 1 and about the order of convexity of functions of the class HS 0 (α, p), (α, p) ∈ A 2 .The next theorems solve this problem.
Proof.The univalence and sense-preservation of the function f in Δ are guaranteed by Theorem 1.
Using a method similar to previously used in [2], J. M. Jahangiri ([6]) has proved that if a function f of the form (1.1) satisfies the condition then for every r ∈ (0, 1), θ ∈ [0, 1], the condition (1.3) holds, so f is starlike of the order β in Δ.
gives an example of such function.
Proof.Univalence and sense-preservation of the function f in Δ are guaranteed by Theorem 1.
We will use the fact (see [6]) that if f of the form (1.1) (b 1 = 0) satisfies the condition then for any r ∈ (0, 1), f satisfies the condition (1.4), so it is a function convex of the order β in the disc Δ.
Using the fact that for any Let us consider a function of the form a(x) = a(α, p; x) : Therefore, we have so, if β = β c (α, p), then the condition (2.4) holds.Therefore, for any function of the class HS 0 (α, p), (α, p) ∈ A 2 and for each β ∈ [0, β c (α, p)], the coefficient condition (2.4) holds.It is a sufficient condition for convexity of the order β of a function f in Δ.

Proposition 2.
In each class HS 0 (α, p), (α, p) ∈ A 2 there exists a function which is convex of the order β * (α, p) in the sense of the restrictive Robertson's definition ( [11]).Formula (2.3) gives an example of such function.
In the proof we use the same method, as in the proof of Proposition 1.  ([11]).
Justification of validity of this property is analogous to the proof of Property 1. HS(α, p) and HS 0 (α, p).The very well-known Alexander theorem for univalent holomorphic functions shows relationships between starlike and convex functions.P. Duren ([4], p. 108) gave the partial extension of this theorem in the case of complex harmonic functions.We show an analogous extension for functions of the classes investigated in this paper.

On other properties of functions of the classes
Indeed, we have The next theorems concern the convolutions of complex harmonic functions in Δ.
Hadamard's convolution of the functions f 1 and f 2 is given by the formula The integral convolution of the functions f 1 and f 2 is given by the formula We have the following result.
If f ∈ HS 0 (α, p), (α, p) ∈ A 2 , then f * f is a univalent sense-preserving function starlike in Δ and f f is a univalent sense-preserving function convex in Δ.
Proof.It is obvious that in each considered case f * f and f f are complex harmonic functions in Δ and have the form required in the class HS 0 (α, p).
Let us assume that (α, p) ∈ dt, z ∈ Δ and f * f ∈ HS 0 (1, 1), we obtain from Property 3 that a function f f belongs to the class HS 0 (1, 2), so it is a univalent and sense-preserving function convex in Δ (see [2]).
Let (α, p) ∈ A 3 .Then it can be shown that U n (α, p) ≥ n 3 , n = 2, 3, . . . .Hence, using an analogous method as above, we can show that for the function f * f the inequalities