On a Subclass of Harmonic Convex Functions of Complex Order

A continuous function f u iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic inΩ. In any simply connected domainD ⊂ Ω, we can write f h g, where h and g are analytic inD. 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 orientation preserving in D is that |h′ z | > |g ′ z | in D see 1 . Denote by SH the family of functions f h g, which are harmonic, univalent, and orientation preserving in the open unit discU {z : |z| < 1} so that f is normalized by f 0 h 0 fz 0 − 1 0. Thus, for f h g ∈ SH, the functions h and g analytic in U can be expressed in the following forms:


Introduction
A continuous function f u iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic in Ω.In any simply connected domain D ⊂ Ω, we can write 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 orientation preserving in Denote by S H the family of functions f h g, which are harmonic, univalent, and orientation preserving in the open unit disc U {z : |z| < 1} so that f is normalized by f 0 h 0 f z 0 − 1 0. Thus, for f h g ∈ S H , the functions h and g analytic in U can be expressed in the following forms: International Journal of Mathematics and Mathematical Sciences and f z is then given by We note that the family S H of orientation preserving, normalized harmonic univalent functions reduces to the well-known class S of normalized univalent functions if the coanalytic part of f is identically zero g ≡ 0 .
Also, we denote by T S H the subfamily of S H consisting of harmonic functions of the form f h g such that h and g are of the form In 1 , Clunie and Sheil-Small investigated the class S H as well as its geometric subclasses and its properties.Since then, there have been several studies related to the class S H and its subclasses.Following Clunie and Sheil-Small 1 , Frasin 2 , Jahangiri et al. 3-6 , Silverman 7 , Silverman and Silvia 8 , Yalc ¸in and Özt ürk 9 , and others have investigated various subclasses of S H and its properties.In particular, Avcı and Złotkiewicz 10 proved that the coefficient condition is sufficient for functions f h g ∈ S H to be harmonic convex.Also, Silverman 7 studied that this coefficient condition is also necessary if a k and b k k 2, 3, . . . in 1.2 are negative.Further, Jahangiri 3 showed that if f h g is given by 1.2 and if then, f is harmonic, univalent, and convex of order γ in U.This condition is proved to be also necessary if h and g are of the form 1.3 .Furthermore, Yalc ¸in and Özt ürk 11 have considered a class T S * H γ of harmonic starlike functions of complex order based on a corresponding study of Nasr and Aouf 12 for the analytic case.
Motivated by the earlier works given in the literature 9, 11 now we define the class of harmonic convex functions of complex order as follows.
We observe that for b 1 the class SC H 1, γ, λ SC H γ, λ was introduced and studied by Yalc ¸in and Özt ürk 9 , the class SC H 1, γ, 0 SC H γ is given in 3, 4 , and the class SC H 1, 0, 0 SC H is studied in 10 .In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for the function class T SC H b, γ, λ .We also examine their convolution and convex combination properties and the closure property of this class under integral operator.We remark that the results obtained for these general families can be viewed as extensions and generalizations for various subclasses of S H as listed previously in this section.

Coefficient Inequalities
Our first theorem gives a sufficient condition for functions in SC H b, γ, λ .Theorem 2.1.Let f h g be so that h and g are given by 1

2.1
where and f is sense preserving, univalent, and harmonic in U.
Proof.We show that f ∈ SC H b, γ, λ .We only need to show that if 2.1 holds, then condition 1.6 is satisfied.In view of 1.2 , condition 1.6 takes the form

2.2
Setting where

2.7
In order to show that f is univalent in U, we show that f z 1 / f z 2 whenever z 1 / z 2 .Since U is simply connected and convex, we have z t 1 − t z 1 tz 2 ∈ U, where 0 ≤ t ≤ 1, and if z 1 , z 2 ∈ U so that z 1 / z 2 .Then, we write Dividing the above equation by z 2 − z 1 / 0 and taking the real part, we obtain

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This along with inequality 2.9 leads to the univalence of f.Note that f is sense preserving in U, for 0 ≤ λ ≤ γ/ 1 γ or λ ≥ 1/ 1 γ .This is because

2.11
The function where shows that the coefficient bound given by 2.1 is sharp.The functions of the form 2.12 are in SC H b, γ, λ because

2.13
The next theorem shows that condition 2.1 is necessary for f ∈ T SC H b, γ, λ .
Theorem 2.2.Let f h g be so that h and g are given by 1.

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Proof.The "if part" follows from Theorem 2.1 upon noting that T SC H b, γ, λ ⊂ SC H b, γ, λ .For the "only if" part, we show that f ∈ T SC H b, γ, λ .Then, for z re iθ in U, we obtain

2.15
The above inequality must hold for all z ∈ U.In particular, letting z r → 1 − yields the required condition.
As special cases of Theorem 2.2, we obtain the following two corollaries.

Corollary 2.4. A function f h g belongs to T SC H b, γ, 1 if and only if
2.17

Extreme Points and Distortion Bounds
In this section, our first theorem gives the extreme points of the closed convex hulls of T SC H b, γ, λ .

International Journal of Mathematics and Mathematical Sciences Theorem 3.1. A function f h g belongs to T SC H b, γ, λ if and only if f can be expressed as
where In particular, the extreme points of T SC H b, γ, λ are {h k } and {g k }.
Proof.For functions f of the form 3.1 , we have where

3.5
The following theorem gives the distortion bounds for functions in T SC H b, γ, λ , which yields a covering result for this family.

3.6
Proof.Let f ∈ T SC H b, γ, λ .Taking the absolute value of f and then by Theorem 2.2, we obtain

International Journal of Mathematics and Mathematical Sciences
Similarly, The upper and lower bounds given in Theorem 3.2 are, respectively, attained for the following functions 3.9

Convolution and Convex Combinations
In this section we show that the class T SC H b, γ, λ is closed under convolution and convex combinations.Now we need the following definition of convolution of two harmonic functions.For we define the convolution of two harmonic functions f and F as Using the definition, we show that the class T SC H b, γ, λ is closed under convolution.Proof.Suppose that f i z ∈ T SC H b, γ, λ , where f i is given by For ∞ i 1 t i 1, 0 ≤ t i ≤ 1, the convex combinations of f i may be written as from the above equation we obtain

4.6
This is the required condition by 2.14 and so ∞ i 1 t i f i z ∈ T SC H b, γ, λ .

Class-Preserving Integral Operator
In this section, we consider the closure property of the class T SC H b, γ, λ under the Bernardi integral operator L c f z , which is defined by

. 2 InternationalTheorem 4 . 2 .
Journal of Mathematics and Mathematical Sciences 11 by Theorem 2.2, f ∈ T SC H b, γ, λ .By the same token, we then conclude that f * F ∈ T SC H b, γ, λ ⊂ T SC H b, δ, λ .Next, we show that the class T SC H b, γ, λ is closed under convex combination of its members.The class T SC H b, γ, λ is closed under convex combinations.