# A Subclass of Harmonic Univalent Functions with Varying Arguments Defined by Generalized Derivative Operator

Authors: E. A. Eljamal; M. Darus
Journal: Advances in Decision Sciences
 (2012)

---

## Abstract

Making use of the generalized derivative operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and extreme points for this generalized class of functions.


---

## Body

## 1. Introduction

A continuous complex-valued functionf=u+iv defined in a simply connected complex domain D is said to be harmonic in D if both u and v are real harmonic in D. Such functions can be expressed as
(1.1)f=h+g¯,
where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h¯(z)|>|g¯(z)| for all z in D (see [1]). Let H be the class of functions of the form (1.1) that are harmonic univalent and sense preserving in the unit disk U={z:|z|<1} for which f(0)=fz(0)-1=0. Then for f=h+g¯∈H, we may express the analytic functions h and g as(1.2)h(z)=z+∑k=2∞akzk,g(z)=z+∑k=1∞bkzk,z∈U,|b1|<1.
In 1984, Clunie and Sheil-Small [1] investigated the class SH as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on SH and its subclasses. Now we will introduce a generalized derivative operator for f=h+g¯ given by (1.2). For fixed positive natural m and λ2≥λ1≥0,(1.3)Dλ1,λ2m,kf(z)=Dλ1,λ2m,kh(z)+Dλ1,λ2m,kg(z)¯,z∈U,
where(1.4)Dλ1,λ2m,kh(z)=z+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))makzk,Dλ1,λ2m,kg(z)=∑k=1∞(1+(λ1+λ2)(k-1)1+λ2(k-1))makzk.
We note that by specializing the parameters, especially when λ1=λ2=0,  Dλ1,λ2m,k reduces to Dm which is introduced by Salagean in [2].Now we will introduce the following definition.Definition 1.1.
For0≤ℓ<1, let GH(ℓ,m,k,λ1,λ2) denote the subfamily of starlike harmonic functions f∈H of the form (1.1) such that
(1.5)Re{(1+eiψ)z(Dλ1,λ2m,kf(z))′z′(Dλ1,λ2m,kf(z))-eiψ}≥l
for a suitable real ψ and z∈U where (Dλ1,λ2m,kf(z))′=(d/dθ)(Dλ1,λ2m,kf(reiθ)),(d/dθ)(z=reiθ).
We also letVH(ℓ,mk,λ1,λ2)=GH(ℓ,mk,λ1,λ2)∩VH where VH is the class of harmonic functions with varying arguments introduced by Jahangiri and Silverman [3] consisting of functions f of the form (1.1) in H for which there exists a real number ϕ such that
(1.6)ηk+(k-1)ϕ≡π(mod⁡2π),δk+(k-1)ϕ≡0(k≥2),
where ηk=arg(ak) and δk=arg(bk). The same class introduced in [4] with different differential operator.
In this paper, we obtain a sufficient coefficient condition for functionsf given by (1.2) to be in the class GH(ℓ,m,k,λ1,λ2). It is shown that this coefficient condition is necessary also for functions belonging to the class VH(ℓ,m,k,λ1,λ2). Further, extreme points for functions in VH(ℓ,m,k,λ1,λ2) are also obtained.


## 2. Main Result

We begin deriving a sufficient coefficient condition for the functions belonging to the classGH(ℓ,m,k,λ1,λ2). This result is contained in the following.Theorem 2.1.
Letf=h+g¯  given by (1.2). Furthermore, let
(2.1)∑k=2∞(2k-1-l1-l|ak|+2k+1+l1-l|bk|)(1+(λ1+λ2)(k-1)1+λ2(k-1))m≤1-3+l3-lb1,
where 0≤ℓ<1, then f∈GH(ℓ,m,k,λ1,λ2).Proof.
We first show that if the inequality (2.1) holds for the coefficients of f=h+g¯, then the required condition (1.5) is satisfied. Using (1.3) and (1.5), we can write
(2.2)Re{(1+eiψ)z(Dλ1,λ2m,kh(z))′-z(Dλ1,λ2m,kg(z))′¯(Dλ1,λ2m,kh(z))+(Dλ1,λ2m,kg(z))¯-eiψ}=ReA(z)B(z),
where
(2.3)A(z)=(1+eiψ)[z(Dλ1,λ2m,kh(z))′-z(Dλ1,λ2m,kg(z))′¯]-eiψ(Dλ1,λ2m,kh(z))+(Dλ1,λ2m,kg(z)),¯B(z)=(Dλ1,λ2m,kh(z))+(Dλ1,λ2m,kg(z)).¯
In view of the simple assertion thatRe(w)≥ℓ if and only if  |1-ℓ+w|≥|1+ℓ-w|, it is sufficies to show that
(2.4)|A(z)+(1-l)B(z)|-|A(z)-(1+l)B(z)|≥0.
Substituting forA(z) and B(z) the appropriate expressions in (2.4), we get
(2.5)|A(z)+(1-l)B(z)|-|A(z)-(1+l)B(z)|≥(2-l)|z|-∑k=2∞(2k-l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak∥z|k-∑k=2∞(2k+l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk∥z|k-l|z|-∑k=2∞(2k-2-l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak∥z|k-∑k=2∞(2k+2+l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk∥z|k≥2(1-l)|z|{1-3+l1-lb1-(∑k=2∞[2k-1-l1-l(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|+2k+1+l1-l(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|2k-1-l1-l(1+(λ1+λ2)(k-1)1+λ2(k-1))m]∑k=2∞)1-3+l1-lb1}∑k=2∞≥0
by virtue of the inequality (2.1). This implies that f∈GH(ℓ,m,k,λ1,λ2).
Now we obtain the necessary and sufficient condition for functionf=h+g¯ be given with condition (1.6).Theorem 2.2.
Letf=h+g¯ be given by (1.2). Then f∈VH(ℓ,m,k,λ1,λ2) if and only if
(2.6)∑k=2∞(2k-1-l1-l|ak|+2k+1+l1-l|bk|)(1+(λ1+λ2)(k-1)1+λ2(k-1))m≤1-3+l3-lb1,
where 0≤ℓ<1.Proof.
SinceVH(ℓ,m,k,λ1,λ2)⊂GH(ℓ,m,k,λ1,λ2), we only need to prove the necessary part of the theorem. Assume that f∈VH(ℓ,m,k,λ1,λ2), then by virtue of (1.3) to (1.5), we obtain
(2.7)Re{(1+eiψ)[z(Dλ1,λ2m,kh(z))′-z(Dλ1,λ2m,kg(z))′¯(Dλ1,λ2m,kh(z))+(Dλ1,λ2m,kg(z))¯-(eiψ+l)]}≥0.
The above inequality is equivalent to(2.8)Re{(1+(λ1+λ2)(k-1)1+λ2(k-1))m(z+(∑k=2∞[k(1+eiψ)-l-eiψ](1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|zk-∑k=2∞[k(1+eiψ)+l+eiψ](1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|zk¯)),×(z+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|zk+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|zk̅)-1}=Re{((1-l)+∑k=2∞[k(1+eiψ)-l-eiψ](1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|zk-1-z¯z∑k=2∞[k(1+eiψ)+l+eiψ](1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|z¯k-1×(1+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|zk-1+z¯z∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|zk-1¯)-1}≥0.
This condition must hold for all values ofz, such that |z|=r<1. Upon choosing ϕ according to (1.6) and noting that Re(-eiψ)≥-|eiψ|=-1, the above inequality reduces to
(2.9)((1-l)-(1-b1)-[∑k=2∞(2k-1-l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|rk-1+(2k+1+l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|rk-1∑k=2∞(2k-1-l)(1+(λ1+λ2)(k-1)1+λ2(k-1))m])×(1-∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|ak|rk-1+∑k=1∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m|bk|rk-1)-1≥0.
If (2.6) does not hold, then the numerator in (2.9) is negative for r sufficiently close to 1. Therefore, there exists a point z0=r0 in (0,1) for which the quotient in (2.9) is negative. This contradicts our assumption that f∈VH(ℓ,m,k,λ1,λ2). We thus conclude that it is both necessary and sufficient that the coefficient bound inequality (2.6) holds true when f∈VH(ℓ,m,k,λ1,λ2). This completes the proof of Theorem 2.2.Theorem 2.3.
The closed convex hull off∈VH(ℓ,m,k,λ1,λ2) (denoted by clco VH(ℓ,m,k,λ1,λ2)) is
(2.10){f(z)=z+∑k=2∞|ak|zk+∑k=1∞|bk|zk¯:∑k=2∞k[|ak|+|bk|]<1-b1}.
By setting λk=(1-ℓ)/((2k-1-ℓ)((1+(λ1+λ2)(k-1))/(1+λ2(k-1)))m) and μk=(1+ℓ)/((2k+1+ℓ)  ((1+(λ1+λ2)(k-1))/(1+λ2(k-1)))m), then for b1 fixed, the extreme points for clco VH(ℓ,m,k,λ1,λ2) are
(2.11){z+λkxzk+b1z¯}∪{z+b1z+μkxzk¯},
where k≥2 and |x|=1-|b1|.Proof.
Any functionf in clco VH(ℓ,m,k,λ1,λ2) may be expressed as
(2.12)f(z)=z+∑k=2∞|ak|eiηkzk+b1z¯+∑k=2∞|bk|eiδkzk¯,
where the coefficients satisfy the inequality (2.1). Set h1(z)=z, g1(z)=b1z, hk(z)=z+λkeiηkzk,  gk(z)=b1z+μkeiδkzk for  k=2,3,…. Writing χk=|ak|/λk,  Yk=|bk|/μk,  k=2,3,… and χ1=1-∑k=2∞χk;  Y1=1-∑k=2∞Yk, we get
(2.13)f(z)=∑k=1∞(χkhk(z)+Ykgk(z)).
In particular, setting
(2.14)f1(z)=z+b1z̅,fk(z)=z+λkxzk+b1z̅+μkyzk̅,(k≥2,|x|+|y|=1-|b1|).
We see that extreme points of clco f∈VH(ℓ,m,kλ1,λ2)⊂{fk(z)}.
To see thatf1(z) is not in extreme point, note that f1(z) may be written as
(2.15)f1(z)=12{f1(z)+λ2(1-|b1|)z2}+12{f1(z)-λ2(1-|b1|)z2},
a convex linear combination of functions in clco VH(ℓ,m,k,λ1,λ2).
To see thatfm is not an extreme point if both |x|≠0 and |y|≠0, we will show that it can then also be expressed as a convex linear combinations of functions in clco VH(ℓ,m,k,λ1,λ2). Without loss of generality, assume |x|≥|y|. Choose ϵ>0 small enough so that ϵ>|x|/|y|. Set A=1+ϵ and B=1-|ϵx/y|. We then see that both(2.16)t1(z)=z+λkAxzk+b1z+μkyBzk¯,t2(z)=z+λk(2-A)xzk+b1z+μky(2-B)zk¯
are in clco VH(ℓ,m,k,λ1,λ2) and that
(2.17)fk(z)=12{t1(z)+t2(z)}.
The extremal coefficient bounds show that functions of the form (2.11) are the extreme points for clco VH(ℓ,m,k,λ1,λ2), and so the proof is complete.Following Avici and Zlotkiewicz [5] and [6], we refer to the δ-neighborhood of the functions f(z) defined by (1.2) to be the set of functions F for which
(2.18)Nδ(f)={F(z)=z+∑k=2∞Akzk+∑k=1∞Bkzk¯,∑k=2∞k(|ak-Ak|+|bk-Bk|+|b1-B1|)≤δ}.In our case, let us define the generalizedδ-neighborhood of f to be the set
(2.19)Nδ(f)={F(z):∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m[(2k-1-l)(|ak-Ak|+(2k+1+l))|bk-Bk|]+(1-l)|b1-B1|≤(1-l)δ∑k=2∞}.Theorem 2.4.
Letf be given by (1.2). If f satisfies the conditions
(2.20)∑k=2∞k(2k-1-l)|ak|(1+(λ1+λ2)(k-1)1+λ2(k-1))m+∑k=1∞k(2k+1+l)|bk|(1+(λ1+λ2)(k-1)1+λ2(k-1))m≤(1-l),
where 0≤ℓ<1, and
(2.21)δ=1-l3-l(1-3+l1-l|b1|),
then N(f)⊂GH(ℓ,m,k,λ1,λ2).Proof.
Letf satisfy (2.20) and F(z) be given by
(2.22)F(z)=z+B1z¯+∑k=2∞(Akzm+Bkzk¯)
which belong to N(f). We obtain
(2.23)(3+l)|B1|+∑k=2∞((2k-1-l)|Ak|+(2k+1+l)|Bk|)(1+(λ1+λ2)(k-1)1+λ2(k-1))m≤(3+l)|B1-b1|+(3+l)|b1|+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m[(2k-1-l)|Ak-ak|+(2k+1+l)|Bk-bk|]+∑k=2∞(1+(λ1+λ2)(k-1)1+λ2(k-1))m[(2k-1-l)|ak|+(2k+1+l)|bk|]≤(1-l)δ+(3+l)|b1|+13-l∑k=2∞k(1+(λ1+λ2)(k-1)1+λ2(k-1))m((2k-1-l)|ak|+(2k+1+l)|bk|)≤(1-l)δ+(3+l)|b1|+13-l[(1-l)-(3+l)|b1|]≤1-l.
Hence for δ=(1-ℓ)/(3-ℓ)(1-((3+ℓ)/(1-ℓ))|b1|), we infer that F(z)∈GH(ℓ,m,k,λ1,λ2) which concludes the proof of Theorem 2.4.



---

*Source: 610406-2012-02-19.xml*