# Subclasses of Harmonic Mappings Defined by Convolution

Authors: Santosh B. Joshi; Girish D. Shelake
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined.


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## Body

## 1. Introduction

A continuous functionf is said to be a complex-valued harmonic function in a simply connected domain D in complex plane ℂ if both real part of f and imaginary part of f are real harmonic in D. Such functions can be expressed as
(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].Every harmonic functionf=h+g- is uniquely determined by the coefficients of power series expansions in the unit disk U={z:|z|<1} given by
(2)h(z)=z+∑n=2∞‍anzn,g(z)=∑n=1∞‍bnzn,z∈U,|b1|<1,
where an∈ℂ for n=2,3,4,… and bn∈ℂ for n=1,2,3,…. For further information about these mappings, one may refer to [1–5].In 1984, Clunie and Sheil-Small [1] studied the family SH of all univalent sense-preserving harmonic functions f of the form (1) in U, such that h and g are represented by (2). Note that SH reduces to the well-known family S, the class of all normalized analytic univalent functions h given in (2), whenever the coanalytic part g of f is zero. Let K and KH denote the respective subclasses of S and SH where the images of f(U) are convex. Denote by SH0 the subclass of SH for which g′(0)=0.The convolution of two functions of the form(3)Φ(z)=z+∑n=2∞‍μnzn,Ψ(z)=z+∑n=2∞‍νnzn
is given by
(4)(Φ*Ψ)(z)=Φ(z)*Ψ(z)=z+∑n=2∞‍μnνnzn,
and the integral convolution is defined by
(5)(Φ⋄Ψ)(z)=Φ(z)⋄Ψ(z)=z+∑n=2∞‍μnνnnzn.
Towards the end of the last century, Jahangiri [3], Silverman [4], and Silverman and Silvia [5] were amongst those who focused on the harmonic starlike functions. Later Öztürk et. al. [6] defined the class SH*(λ,α) consisting of functions f=h+g- such that h and g are of the forms
(6)h(z)=z-∑n=2∞‍|an|zn,g(z)=∑n=1∞‍|bn|zn,
which satisfy the condition
(7)Re{zh′(z)-zg′(z)¯λ(zh′(z)-zg′(z)¯)+(1-λ)(h(z)+g(z)¯)}>α,
for some 0≤α<1,0≤λ≤1 and for all z∈U.Several authors [3–16] have investigated various subclasses of harmonic functions. In this work, we introduce a new subclass of harmonic functions defined by convolution.Letσ be a real constant with |σ|=1, then we denote SH0(Φ,Ψ,σ,λ,α), the subclass of SH0 of functions of the form f=h+g-∈SH0 that satisfy the condition
(8)Re{H(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)}>α,
where H(f,Φ,Ψ)=h(z)*Φ(z)-σg(z)*Ψ(z)¯, G(f,Φ,Ψ)=h(z)⋄Φ(z)+σg(z)⋄Ψ(z)¯, 0≤α<1, 0≤λ≤1, and Φ,Ψ are as given in (3).We also denoteSPH0(Φ,Ψ,σ,λ,α), the subclass of SH0 of functions of the form f=h+g-∈SH0 that satisfy the condition
(9)Re{(1+eiγ)H(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)-eiγH(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)}>α,
where γ is real.We note that the familiesSH0(Φ,Ψ,σ,λ,α) and SPH0(Φ,Ψ,σ,λ,α) are of special interest, because they contain various classes of well-known harmonic univalent functions as well as many new ones. For different choice of Φ,Ψ,λ, and α we obtain the following various classes introduced by other authors:(1)
SH0((z/(1-z)2),(z/(1-z)2),1,λ,α)=SH0(λ,α) (see Öztürk et. al [6]).(2)
SH0((z/(1-z)2),(z/(1-z)2),1,0,α)=SH0(α) (see Jahangiri [3]).(3)
SH0((z/(1-z)2),(z/(1-z)2),1,0,0)=SH0 (see Silverman and Silvia [5]).(4)
SH0((z/(1-z)2),(z/(1-z)2),1,0,0)=SH*0, with b1=0 (see Avcı and Złotkiewicz [17] and Silverman [4]).(5)
SH0(((z+z2)/(1-z)3),((z+z2)/(1-z)3),1,0,α)=KH0(α) (see Jahangiri [3]).(6)
SH0(((z+z2)/(1-z)3),((z+z2)/(1-z)3),1,0,0)=KH*0 (see Silverman [4]).(7)
SH0(Φ,Ψ,1,0,α)=SH0(Φ,Ψ,α) (see Dixit et al. [11]).(8)
SH0(Φ,Φ,σ,0,α)=SH0(Φ,Φ,α) (see Ali et al. [7]).(9)
SH0(Φ,Ψ,1,λ,α)=SH0(Φ,Ψ,λ,α) (see Joshi et al. [14]).(10)
SH0(Φ,Ψ,(1)l,0,α)=H(l,α), where μn=νn=nl (see Jahangiri et al. [13]).(11)
SH0((z/(1-z)γ+1),(z/(1-z)γ+1),1,0,α)=RH(γ,α), where γ>-1 (see Murugusundaramoorthy [15]).(12)
SH0(Φ,Ψ,(-1)l,0,α)=MH(l,γ,α), where μn=νn=nl(((γ+1)n-1)/(n-1)!) and (γ+1)n-1=(γ+1)(γ+2)⋯(γ+n-1) (see Al-Shaqsi and Darus [10]).(13)
SH0(Φ,Ψ,1,λ,α)=RHp,q(Φ,Ψ,λ,α), where μn=νn=(((γ1)n-1…(γp)n-1)/((β1)n-1…(βp)n-1))(1/(n-1)!) (see Murugusundaramoorthy et al. [16]).It is clear that the classSH0(Φ,Ψ,σ,λ,α) generates a number of known subclasses and provides a unified treatment of these subclasses of harmonic mappings. Motivated by work of Ali et al. [7], we obtain convolution characterization for functions in the class SH0(Φ,Ψ,σ,λ,α) and SPH0(Φ,Ψ,σ,λ,α). We also obtain sufficient coefficient condition for these two classes, and the last section is devoted to determine growth estimates and extreme points for the class TSH0(Φ,Ψ,σ,λ,α) and TSPH0(Φ,Ψ,σ,λ,α).


## 2. Main Results

We now derive a convolution characterization for functions in the classSH0(Φ,Ψ,σ,λ,α).Theorem 1.
Letf=h+g-∈SH0. Then f∈SH0(Φ,Ψ,σ,λ,α) if and only if
(10)(h*Φ)⋄[z+(((x+2α-1)(1-λ))/(2-2α))z2(1-z)2]-σ(g*Ψ)¯⋄[(x+α-λ(x+2α-1)1-αz¯-(x+2α-1)(1-λ)2-2αz¯2)((1-z¯)2)-1(x+α-λ(x+2α-1)1-αz¯-(x+2α-1)(1-λ)2-2αz¯2)]≠0,|x|=1,|z|≠0.Proof.
A necessary and sufficient condition forf=h+g- to be in the class SH0(Φ,Ψ,σ,λ,α), with h and g of the form (1) is given by (8). The condition (8) holds if and only if
(11)1(1-α){H(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)-α}≠x-1x+1;|x|=1,x≠-1,0<|z|<1.
By simple algebraic manipulation, (11) gives
(12)0≠(x+1)[h(z)*Φ(z)-σg(z)*Ψ(z)¯]-(x+2α-1)[λ(h(z)*Φ(z)-σg(z)*Ψ(z)¯)+(1-λ)(h(z)⋄Φ(z)+σg(z)⋄Ψ(z)¯)]=(h*Φ)⋄[2(1-α)z+(x-1+2α)(1-λ)z2(1-z)2]-σ(g*Ψ)¯⋄[2(x¯+α-λ(x¯+2α-1))z-(x¯+2α-1)(1-λ)z2(1-z)2]¯.
The latter condition, along with (8) for x=-1, establishes the result for all |x|=1.An application of the convolution condition in Theorem1 gives sufficient condition for harmonic functions to belong to the class SH0(Φ,Ψ,σ,λ,α).Theorem 2.
Letf=h+g-∈SH0. Then f∈SH0(Φ,Ψ,σ,λ,α) if
(13)∑n=2∞‍(n-α-λα(n-1)1-α)|μn||an|n+∑n=2∞‍(n+α-λα(n+1)1-α)|νn||bn|n≤1.Proof.
Forh and g given by (2), Theorem 1 gives
(14)|(h*Φ)⋄[(z+(x+2α-1)(1-λ)2-2αz2)×((1-z)2)-1(x+2α-1)(1-λ)2-2α]-σ(g*Ψ)¯⋄[x+α-λ(x+2α-1)1-αz¯-(x+2α-1)(1-λ)2-2αz¯2×((1-z¯)2)-1x+α-λ(x+2α-1)1-α](x+2α-1)(1-λ)2-2α|=|z+∑n=2∞‍μnn[n+(n-1)(x+2α-1)(1-λ)2-2α]anzn-σ∑n=2∞‍νnn[nx+α-λ(x+2α-1)1-α-(n-1)(x+2α-1)(1-λ)2-2α]bnzn¯|>|z|[1-∑n=2∞‍(n-α-λα(n-1)1-α)|μn||an|n-∑n=2∞‍(n+α-λα(n+1)1-α)|νn||bn|n].
The last expression is nonnegative by hypothesis, and hence by Theorem 1, it follows that f∈SH0(Φ,Ψ,σ,λ,α).Theorem 3.
Letf=h+g-∈SH0. Then f∈SPH0(Φ,Ψ,σ,λ,α) if and only if
(15)(h*Φ)⋄[z+[(x+1)eiγ+x+2α-1](1-λ)2-2αz2×((1-z)2)-1[(x+1)eiγ+x+2α-1](1-λ)2-2α]-σ(g*Ψ)¯⋄[((x+1)eiγ+x+α-λ((x+1)eiγ+x+2α-1)1-αz¯-[(x+1)eiγ+x+2α-1](1-λ)2-2αz¯2)((1-z¯)2)-1[(x+1)eiγ+x+2α-1](1-λ)2-2α]≠0,|x|=1,|z|≠0.Proof.
A necessary and sufficient condition forf=h+g- to be in the class SH0(Φ,Ψ,σ,λ,α), with h and g of the form (1) is given by (9). The condition (9) holds if and only if
(16)1(1-α){(1+eiγ)H(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)-eiγ-αH(f,Φ,Ψ)λH(f,Φ,Ψ)+(1-λ)G(f,Φ,Ψ)}≠x-1x+1;|x|=1,x≠-1,0<|z|<1.
By simple algebraic manipulation, we get the desired result.Now sufficient coefficient condition for the classSPH0(Φ,Ψ,σ,λ,α) is easily obtained.Theorem 4.
Letf=h+g-∈SH0. Then f∈SPH0(Φ,Ψ,σ,λ,α) if
(17)∑n=2∞‍(2n-1-α-λ(1+α)(n-1)1-α)|μn||an|n+∑n=2∞‍(2n+1+α-λ(1+α)(n+1)1-α)|νn||bn|n≤1.We further letTSH0(Φ,Ψ,σ,λ,α) and TSPH0(Φ,Ψ,σ,λ,α) denote the subclasses of SH0(Φ,Ψ,σ,λ,α) and SPH0(Φ,Ψ,σ,λ,α), respectively, consisting of functions f=h+g-∈SH0 such that h and g are of the form
(18)h(z)=z-∑n=2∞‍anzn,g(z)=σ∑n=2∞‍bnzn,an≥0,bn≥0.
Let
(19)Φ(z)=z+∑n=2∞‍μnzn,Ψ(z)=z+∑n=2∞‍νnzn,
with μn≥0,νn≥0.Theorem 5.
Letf=h+g- of the form (18). Then f∈TSH0(Φ,Ψ,σ,λ,α) if and only if
(20)∑n=2∞‍μnn(n-α-λα(n-1)1-α)an+∑n=2∞‍νnn(n+α-λα(n+1)1-α)bn≤1.Proof.
Iff∈TSH0(Φ,Ψ,σ,λ,α), then (8) is equivalent to
(21)Re{((1-α)z-∑n=2∞‍μn[n-α-αλ(n-1)]nanzn-∑n=2∞‍νn[n+α-αλ(n+1)]nbnz-n)×(z-∑n=2∞‍μn[λ+(1-λn)]anzn+∑n=2∞‍νn[(1-λn)-λ]bnz-n)-1}>0,
for z∈U. Letting z→1- through real values, we get
(22)((1-α)-∑n=2∞‍μn([n-α-αλ(n-1)]n)an-∑n=2∞‍νn([n+α-αλ(n+1)]n)bn)×(1-∑n=2∞‍μn[λ+(1-λn)]an+∑n=2∞‍νn[(1-λn)-λ]bn)-1≥0,
which gives required condition (20).
Conversely, forh and g given by (18),
(23)|(h*Φ)⋄[(z+(x+2α-1)(1-λ)2-2αz2)×((1-z)2)-1(x+2α-1)(1-λ)2-2α]-σ(g*Ψ)¯⋄[(×((1-z¯)-2)-1x+α-λ(x+2α-1)1-αz¯-(x+2α-1)(1-λ)2-2αz¯2)×((1-z¯)2)-1(x+2α-1)(1-λ)2-2α]|>|z|[1-∑n=2∞‍μnn(n-α-λα(n-1)1-α)an-∑n=2∞‍νnn(n+α-λα(n+1)1-α)bn],
which is nonnegative by hypothesis.Theorem 6.
Letf=h+g- of the form (18). Then f∈TSPH0(Φ,Ψ,σ,λ,α) if and only if
(24)∑n=2∞‍μnn(2n-1-α-λ(1+α)(n-1)1-α)an+∑n=2∞‍νnn(2n+1+α-λ(1+α)(n+1)1-α)bn≤1.The following theorem gives the distortion bounds for functions inTSH0(Φ,Ψ,σ,λ,α), and TSPH0(Φ,Ψ,σ,λ,α) which gives a covering result for the classes TSH0(Φ,Ψ,σ,λ,α) and TSPH0(Φ,Ψ,σ,λ,α), respectively.Theorem 7.
Letf∈TSH0(Φ,Ψ,σ,λ,α) and μn≥ν2,νn≥ν2,(n≥2), then for |z|=r<1, we have
(25)r-2(1-α)(2-α-λα)ν2r2≤|f(z)|≤r+2(1-α)(2-α-λα)ν2r2.
The result is sharp with equality for f(z)=z-(2(1-α)/(2-α-λα)ν2)z2.Proof.
We have(26)(2-α-αλ)ν22∑n=2∞‍(an+bn)≤∑n=2∞‍[n-α-λα(n-1)]μnann+∑n=2∞‍[n+α-λα(n+1)]νnbnn≤1-α.
Thus,
(27)|f(z)|=|z-∑n=2∞‍anzn+σ∑n=2∞‍bnz¯n|≤r+r2∑n=2∞‍(an+bn)≤r+2(1-α)(2-α-λα)ν2r2.
The proof of left-hand inequality follows in lines similar to that of right-hand side inequality.Theorem 8.
Letf∈TSPH0(Φ,Ψ,σ,λ,α) and μn≥ν2,νn≥ν2,(n≥2), then for |z|=r<1, we have
(28)r-2(1-α)(3-α-λ(1+α))ν2r2≤|f(z)|≤r+2(1-α)(3-α-λ(1+α))ν2r2.
The result is sharp with equality for f(z)=z-(2(1-α)/(3-α-λ(1+α))ν2)z2.Corollary 9.
Letf∈TSH0(Φ,Ψ,σ,λ,α), then
(29){w:|w|<1-2(1-α)(2-α-λα)ν2}⊂f(U).Corollary 10.
Letf∈TSPH0(Φ,Ψ,σ,λ,α), then
(30){w:|w|<1-2(1-α)(3-α-λ(1+α))ν2}⊂f(U).Finally, we determine the extreme points of the classTSH0(Φ,Ψ,σ,λ,α), and TSPH0(Φ,Ψ,σ,λ,α).Theorem 11.
Let(31)h1(z)=z,hn(z)=z-n(1-α)[n-α-λα(n-1)]μnzn,gn(z)=z-n(1-α)σ[n+α-λα(n+1)]νnz¯n(n=2,3,…).
A function f∈TSH0(Φ,Ψ,σ,λ,α) if and only if f can be expressed in the form
(32)f(z)=∑n=1∞‍(βnhn+γngn),
where βn≥0, γn≥0, β1=1-∑n=2∞‍(βn+γn) and γ1=0. In particular, the extreme points of TSH0(Φ,Ψ,σ,λ,α) are {hn} and {gn}.Proof.
Let(33)f(z)=∑n=2∞‍(βnhn+γngn)=z-∑n=2∞‍βnn(1-α)[n-α-λα(n-1)]μnzn+σ∑n=2∞‍γnn(1-α)[n+α-λα(n+1)]νnz¯n.
Since
(34)∑n=2∞‍μnn(n-α-λα(n-1)1-α)βnn(1-α)[n-α-λα(n-1)]μn+∑n=1∞‍νnn(n+α-λα(n+1)1-α)γnn(1-α)[n+α-λα(n+1)]νn=∑n=2∞‍(βn+γn)=1-β1≤1,
from Theorem 5, f∈TSH0(Φ,Ψ,σ,λ,α).
Conversely, iff∈TSH0(Φ,Ψ,σ,λ,α), then  an≤n(1-α)/(n-α-λα(n-1))μn   and  bn≤n(1-α)/(n+α-λα(n+1)νn). Set βn=(μn/n)(n-α-λα(n-1)/(1-α))an, γn=(νn/n)(n+α-λα(n+1)/(1-α))bn, β1=1-∑n=2∞‍(βn+γn), and γ1=0.
Then,(35)∑n=2∞‍(βnhn+γngn)=z-∑n=2∞‍anzn+σ∑n=2∞‍bnz¯n=f(z).Theorem 12.
Let(36)p1(z)=z,pn(z)=z-n(1-α)[2n-1-α-λ(1+α)(n-1)]μnzn,qn(z)=z-n(1-α)σ[2n+1+α-λ(1+α)(n+1)]νnz¯n(n=2,3,…).
A function f∈TSPH0(Φ,Ψ,σ,λ,α) if and only if f can be expressed in the form
(37)f(z)=∑n=1∞‍(βnpn+γnqn),
where βn≥0, γn≥0, β1=1-∑n=2∞‍(βn+γn), and γ1=0. In particular, the extreme points of TSPH0(Φ,Ψ,σ,λ,α) are {pn} and {qn}.



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