A new subclass of the meromorphic harmonic γ-starlike functions
Introduction
A continuous function f = u + iv is a complex valued harmonic function in a complex domain D if both u and v are real harmonic in D. In any simply connected domain , we can write , where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that ∣h′(z)∣ > ∣g′(z)∣ in D (see [3]). There are numerous papers on univalent harmonic functions defined on the domain (see [3], [1], [5], [6]). In [4] Hengartner and Schober investigated functions harmonic in the exterior of the unit disc . They showed that a complex valued, harmonic, sense preserving, univalent mapping f, defined on and satisfying f(∞) = ∞, must admit the representationwhere and is analytic and satisfies ∣a(z)∣ < 1 for . After this work, Jahangiri and Silverman [7], gave sufficient coefficient conditions for which functions of the form (1) will be univalent. Under certain restrictions, they also gave necessary and sufficient coefficient conditions for functions to be harmonic and starlike. In [7], the following theorem, which we shall use in this work, is also proved. Theorem 1.1 Let where h(z) and g(z) are of the form Eq. (2). Ifthen f(z) is sense preserving and univalent in .
Recently, Bostanci and Öztürk [2], have defined the following operator Mn for meromorphic harmonic functions where h and g are of the form (2), as follows:and for n = 2, …Hence, they obtain for n = 0, 1, …In [2], they defined the classes MH∗(n) and . Also they investigated some properties of these classes such as coefficient estimates and distortion theorems.
In this work, we extended this results to the general cases MH∗(n, γ) and . Using the operator Mn, we now introduce the following classes:
Let MH∗(n, γ) denote the class of meromorphic harmonic, sense preserving, univalent functions that satisfy the following condition.Also, let be the subclass of MH∗(n, γ) which consists of meromorphic harmonic functions of the formwhere α > 3nβ ⩾ 0, ak ⩾ 0, bk ⩾ 0 and b2 ⩽ (α − β)/2.
For 0 ⩽ γ1 < γ2 < (α + 3nβ)/(α − 3nβ), MH∗(n, γ2) ⊂ MH∗(n, γ1) ⊂ MH∗(n, 0) and .
Notice that if we take n = 0 in the inequality Eq. (4), then we obtainwhere z = reiθ, 0 ⩽ θ < 2π, r > 1. The inequality (6) is a necessary and sufficient condition for functions f of the form (1) to be γ-starlike in . This classification in (6) for harmonic univalent functions was first used by Jahangiri, [8].
Also, specializing the parameters n, α, β and γ, we have.
- (i)
MH∗(n, 0) = MH∗(n) (see [2]);
- (ii)
denotes the subclass of harmonic sense preserving functions f that are starlike in and (see [7]);
- (iii)
If we substitute α = 1, β = 0, then we have , (see [9]).
Section snippets
Coefficient inequalities
In this section we obtain coefficient bounds. Our first theorem gives a sufficient coefficient condition for the class MH∗(n, γ). Theorem 2.1 If where ∣b2∣ ⩽ (∣α∣ − ∣β∣)/2, h(z) and g(z) of the form (2) and the conditionis satisfied then f(z) is univalent, sense preserving in and f(z) ∈ MH∗(n, γ). Proof In view of Theorem 1.1, f(z) is sense preserving and univalent in . Now it remains to show that the condition (7) is sufficient for fto
Convex combinations
In this section, we show that the class is invariant under convex combinations of its members. Theorem 3.1 The class is a convex set. Proof For i = 1, 2, … suppose that where fn,i(z) is given byThen, by Theorem 2.2,For , the convex combination of fn,i may be written as
A distortion theorem and extreme points
Theorem 4.1 Let the function fn(z) be in the class . Then, for ∣z∣ = r > 1, we have Proof Let . Taking the absolute value of fn(z), we obtainThe proof for the bound on the right hand is similar to that given above and we omit it. □ Corollary 4.2 Let 0 ⩽ γ < (α + 3nβ)/(α − 3nβ) and . Then
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