A new subclass of the meromorphic harmonic γ-starlike functions

Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary
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Abstract

Recently Bostancı and Öztürk defined a new operator Mn for meromorphic harmonic functions. They introduced new classes of meromorphic harmonic starlike functions in U={z:|z|>1} using operator Mn. In this work, we have generalized these classes to meromorphic harmonic γ-starlike functions. Also, we have examined some properties of these classes.

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 DC, we can write f=h+g¯, 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 U={z:|z|<1} (see [3], [1], [5], [6]). In [4] Hengartner and Schober investigated functions harmonic in the exterior of the unit disc U={z:|z|>1}. They showed that a complex valued, harmonic, sense preserving, univalent mapping f, defined on U and satisfying f(∞) = ∞, must admit the representationf(z)=h(z)+g(z)¯+Alog|z|,whereh(z)=αz+k=1akz-kandg(z)=βz+k=1bkz-k.0|β|<|α|,AC and a=f¯z¯/fz is analytic and satisfies ∣a(z)∣ < 1 for zU. 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 f(z)=h(z)+g(z)¯+Alog|z| where h(z) and g(z) are of the form Eq. (2). Ifk=1k(|ak|+|bk|)|α|-|β|-|A|,then f(z) is sense preserving and univalent in U.

Recently, Bostanci and Öztürk [2], have defined the following operator Mn for meromorphic harmonic functions f=h+g¯ where h and g are of the form (2), as follows:M0f(z)=f(z),M1f(z)=Mf(z)=(z2g(z))z¯-z3h(z)z2and for n = 2, Mnf(z)=M(Mn-1f(z)).Hence, they obtain for n = 0, 1, Mnf(z)=αz+k=1(k+2)nakz-k+3nβz+(-1)nk=1(k-2)nbkz-k¯.In [2], they defined the classes MH(n) and MH¯(n). 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 MH¯(n,γ),0γ<(α+3nβ)/(α-3nβ). 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.Re2-Mn+1f(z)Mnf(z)>γ,zU,nN0={0,1,2,}.Also, let MH¯(n,γ) be the subclass of MH(n, γ) which consists of meromorphic harmonic functions of the formfn(z)=h(z)+gn(z)¯=-αz-k=1akz-k+βz-(-1)nk=1bkz-k¯,where 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 MH¯(n,γ2)MH¯(n,γ1)MH¯(n,0).

Notice that if we take n = 0 in the inequality Eq. (4), then we obtainγ<Rezh(z)-zg(z)¯h(z)+g(z)¯=Imθlogf(reiθ)=θargf(reiθ),where z = re, 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 U. 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)

    MH(0,0)=H0 denotes the subclass of harmonic sense preserving functions f that are starlike in U and MH¯(0,0)=TH0 (see [7]);

  • (iii)

    If we substitute α = 1, β = 0, then we have MH(0,0)=ΣH(0), (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 f(z)=h(z)+g(z)¯ whereb2  (α  β)/2, h(z) and g(z) of the form (2) and the conditionk=1(k+γ)(k+2)n|ak|+k=3(k-γ)(k-2)n|bk|+|b1|(1-γ)|α|-(1+γ)3n|β|is satisfied then f(z) is univalent, sense preserving in U and f(z)  MH(n, γ).

Proof

In view of Theorem 1.1, f(z) is sense preserving and univalent in U. Now it remains to show that the condition (7) is sufficient for fto

Convex combinations

In this section, we show that the class MH¯(n,γ) is invariant under convex combinations of its members.

Theorem 3.1

The class MH¯(n,γ) is a convex set.

Proof

For i = 1, 2, … suppose that fn,i(z)MH¯(n,γ) where fn,i(z) is given byfn,i(z)=-αiz-k=1ak,iz-k+βiz-(-1)nk=1bk,iz-k¯.Then, by Theorem 2.2,k=1(k+γ)(k+2)nak,i+k=3(k-γ)(k-2)nbk,i+(1-γ)b1,i(1-γ)αi-(1+γ)3nβi.For i=1ti=1,0ti1, the convex combination of fn,i may be written asi=1tifn,i(z)=-i=1tiαiz-k=1i=1tiak,iz-k+i=1tiβiz¯-(-1)nk=1i=1tibk,iz¯

A distortion theorem and extreme points

Theorem 4.1

Let the function fn(z) be in the class MH¯(n,γ). Then, forz = r > 1, we have(α-β)r-(1-γ)α-(1+γ)3nβr-1|fn(z)|(α+β)r+(1-γ)α-(1+γ)3nβr-1.

Proof

Let fn(z)MH¯(n,γ). Taking the absolute value of fn(z), we obtain|fn(z)|=-αz-k=1akz-k+βz-(-1)nk=1bkz-k¯αr+βr+k=1(ak+bk)r-kαr+βr+k=1(ak+bk)r-1αr+βr+r-1k=1(k+γ)(k+2)nak+k=3(k-γ)(k-2)nbk+(1-γ)b1(α+β)r+(1-γ)α-(1+γ)3nβr-1.The proof for the bound on the right hand is similar to that given above and we omit it. 

Corollary 4.2

Let 0  γ <  + 3nβ)/(α  3nβ) and nN0. Then MH

References (9)

  • Y. Avci et al.

    On harmonic univalent mappings

    Ann. Univ. Mariae Curie-Sklodowska Sect. A

    (1990)
  • H. Bostanci et al.

    A new subclass of the meromorphic harmonic starlike functions

    Appl. Math. Lett.

    (2010)
  • J. Cluine et al.

    Harmonic univalent functions

    Ann. Acad. Sci. Fenn. Ser. Al Math.

    (1984)
  • W. Hengartner et al.

    Univalent harmonic functions

    Trans. Amer. Math. Soc.

    (1987)
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