# Coefficient Conditions for Harmonic Close-to-Convex Functions

Authors: Toshio Hayami
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

New sufficient conditions, concerned with the coefficients of harmonic functionsf(z)=h(z)+g(z)¯ in the open unit disk 𝕌 normalized by f(0)=h(0)=h′(0)−1=0, for f(z) to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domains of harmonic close-to-convex functions satisfying the obtained conditions are enumerated.


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## Body

## 1. Introduction

For a continuous complex-valued functionf(z)=u(x,y)+iv(x,y)(z=x+iy), we say that f(z) is harmonic in the open unit disk 𝕌={z∈ℂ:|z|<1} if both u(x,y) and v(x,y) are real harmonic in 𝕌, that is, u(x,y) and v(x,y) satisfy the Laplace equations(1.1)Δu=uxx+uyy=0,Δv=vxx+vyy=0.
A complex-valued harmonic function f(z) in 𝕌 is given by f(z)=h(z)+g(z)¯ where h(z) and g(z) are analytic in 𝕌. We call h(z) and g(z) the analytic part and the coanalytic part of f(z), respectively. A necessary and sufficient condition for f(z) to be locally univalent and sense preserving in 𝕌 is |h′(z)|>|g′(z)| in 𝕌 (see [1] or [2]). Let ℋ denote the class of harmonic functions f(z) in 𝕌 with f(0)=h(0)=0 and h′(0)=1. Thus, every normalized harmonic function f(z) can be written by(1.2)f(z)=h(z)+g(z)¯=z+∑n=2∞anzn+∑n=1∞bnzn¯∈H,
where a1=1 and b0=0, for convenience.We next denote by𝒮ℋ the class of functions f(z)∈ℋ that are univalent and sense preserving in 𝕌. Due to the sense-preserving property of f(z), we see that |b1|=|g′(0)|<|h′(0)|=1. If g(z)≡0, then 𝒮ℋ reduces to the class 𝒮 consisting of normalized analytic univalent functions. Furthermore, for every function f(z)∈𝒮ℋ, the function(1.3)F(z)=f(z)-b1f(z)¯1-|b1|2=z+∑n=2∞an-b1¯bn1-|b1|2zn+∑n=2∞bn-b1an1-|b1|2zn¯
is also a member of 𝒮ℋ. Therefore, we consider the subclass 𝒮ℋ0 of 𝒮ℋ defined as(1.4)SH0={f(z)∈SH:b1=g′(0)=0}.
Conversely, if F(z)∈𝒮ℋ0, then f(z)=F(z)+b1F(z)¯∈𝒮ℋ for any b1(|b1|<1).We say that a domain𝔻 is a close-to-convex domain if the complement of 𝔻 can be written as a union of nonintersecting half-lines (except that the origin of one half-line may lie on one of the other half-lines). Let 𝒞, 𝒞ℋ, and 𝒞ℋ0 be the respective subclasses of 𝒮, 𝒮ℋ, and 𝒮ℋ0 consisting of all functions f(z), which map 𝕌 onto a certain close-to-convex domain.Bshouty and Lyzzaik [3] have stated the following result.Theorem 1.1.
Iff(z)=h(z)+g(z)¯∈ℋ satisfies
(1.5)g′(z)=zh′(z),Re(1+zh″(z)h′(z))>-12
for all z∈𝕌, then f(z)∈𝒞ℋ0⊂𝒮ℋ0.A simple and interesting example is below.Example 1.2.
The function(1.6)f(z)=1-(1-z)22(1-z)2+z22(1-z)2¯=z+∑n=2∞n+12zn+∑n=2∞n-12z¯n
satisfies the conditions of Theorem 1.1, and therefore f(z) belongs to the class 𝒞ℋ0. We now show that f(𝕌) is actually a close-to-convex domain. It follows that
(1.7)f(z)=(z2(1-z)2+z2(1-z))+(z2(1-z)2-z2(1-z))¯=Re(z(1-z)2)+iIm⁡(z1-z).
Setting
(1.8)f(reiθ)=-2r2+r(1+r2)cos⁡θ(1+r2-2rcos⁡θ)2+rsin⁡θ1+r2-2rcos⁡θi=u+iv
for any z=reiθ∈𝕌(0≦r<1,0≦θ<2π), we see that
(1.9)-4(u+v2)=4r(r-cos⁡θ)(1-rcos⁡θ)(1+r2-2rcos⁡θ)2=4r(r-t)(1-rt)(1+r2-2rt)2≡ϕ(t)(-1≦t=cos⁡θ≦1).
Since
(1.10)ϕ′(t)=-4r(1-r2)2(1+r2-2rt)3≦0,
we obtain that
(1.11)ϕ(t)≦ϕ(-1)=4r(1+r)2≡ψ(r).
Also, noting that
(1.12)ψ′(r)=4(1-r)(1+r)3>0,
we know that
(1.13)ψ(r)<ψ(1)=1,
which implies that
(1.14)u>-v2-14.
Thus, f(z) maps 𝕌 onto the following close-to-convex domain as shown in Figure 1.Figure 1
The image off(z)=(1-(1-z)2)/2(1-z)2+z2/2(1-z)2¯.Remark 1.3.
Letℳ be the class of all functions satisfying the conditions of Theorem 1.1. Then, it was earlier conjectured by Mocanu [4, 5] that ℳ⊂𝒮ℋ0. Furthermore, we can immediately see that the function f(z) in Example 1.2 is a member of the class ℳ and it shows that f(z)∈ℳ is not necessarily starlike with respect to the origin in 𝕌 (f(z) is starlike with respect to the origin in 𝕌 if and only if tw∈f(𝕌) for all w∈f(𝕌) and t(0≦t≦1)).Remark 1.4.
For the functionf(z)=h(z)+g(z)¯∈ℋ given by
(1.15)g′(z)=zn-1h′(z)(n=2,3,4,…),
letting w(t)=f(eit)=h(eit)+g(eit)¯(-π≦t<π), we know that
(1.16)Im⁡(w′′(t)w′(t))≦0(-π≦t<π),
which means that f(z) maps the unit circle ∂𝕌={z∈ℂ:|z|=1} onto a union of several concave curves (see [6, Theorem 2.1]).
Jahangiri and Silverman [7] have given the following coefficient inequality for f(z)∈ℋ to be in the class 𝒞ℋ.Theorem 1.5.
Iff(z)∈ℋ satisfies
(1.17)∑n=2∞n∣an∣+∑n=1∞n∣bn∣≦1,
then f(z)∈𝒞ℋ.Example 1.6.
The function(1.18)f(z)=z+15z¯5
belongs to the class 𝒞ℋ0⊂𝒞ℋ and satisfies the condition of Theorem 1.5. Indeed, f(z) maps 𝕌 onto the following hypocycloid of six cusps (cf. [8] or [6]) as shown in Figure 2.
The object of this paper is to find some sufficient conditions for functionsf(z)∈ℋ to be in the class 𝒞ℋ. In order to establish our results, we have to recall here the following lemmas due to Clunie and Sheil-Small [1].Figure 2
The image off(z)=z+(1/5)z¯5.Lemma 1.7.
Ifh(z) and  g(z) are analytic in  𝕌 with  |h′(0)|>|g′(0)| and  h(z)+ɛg(z) is close-to-convex for each  ɛ(|ɛ|=1), then  f(z)=h(z)+g(z)¯ is harmonic close-to-convex.Lemma 1.8.
Iff(z)=h(z)+g(z)¯ is locally univalent in  𝕌 and  h(z)+ɛg(z) is convex for some  ɛ(|ɛ|≦1), then  f(z) is univalent close-to-convex.We also need the following result due to Hayami et al. [9].Lemma 1.9.
If a functionF(z)=z+∑n=2∞Anzn is analytic in  𝕌 and satisfies
(1.19)∑n=2∞[|∑k=1n{∑j=1k(-1)k-jj(j+1)(αk-j)Aj}(βn-k)|+|∑k=1n{∑j=1k(-1)k-jj(j-1)(αk-j)Aj}(βn-k)|]≦2
for some real numbers α and β, then F(z) is convex in 𝕌.


## 2. Main Results

Our first result is contained in the following theorem.Theorem 2.1.
Iff(z)∈ℋ satisfies the following condition
(2.1)∑n=2∞|nan-eiφ(n-1)an-1|+∑n=1∞|nbn-eiφ(n-1)bn-1|≦1
for some real number φ(0≦φ<2π), then f(z)∈𝒞ℋ.Proof.
LetF(z)=z+∑n=2∞Anzn be analytic in 𝕌. If F(z) satisfies
(2.2)∑n=2∞|nAn-eiφ(n-1)An-1|≦1,
then it follows that
(2.3)|(1-eiφz)F′(z)-1|=|∑n=2∞(nAn-eiφ(n-1)An-1)zn-1|≦∑n=2∞|nAn-eiφ(n-1)An-1|⋅∣z∣n-1<∑n=2∞|nAn-eiφ(n-1)An-1|≦1(z∈U).
This gives us that
(2.4)Re((1-eiφz)F′(z))>0(z∈U),
that is, F(z)∈𝒞. Then, it is sufficient to prove that
(2.5)F(z)=h(z)+ɛg(z)1+ɛb1=z+∑n=2∞an+ɛbn1+ɛb1zn∈C
for each ɛ(|ɛ|=1) by Lemma 1.7. From the assumption of the theorem, we obtain that
(2.6)∑n=2∞|nan+ɛbn1+ɛb1-eiφ(n-1)an-1+ɛbn-11+ɛb1|≦11-∣b1∣∑n=2∞[|nan-eiφ(n-1)an-1|+|nbn-eiφ(n-1)bn-1|]≦1-∣b1∣1-∣b1∣=1.
This completes the proof of the theorem.Example 2.2.
The function(2.7)f(z)=-log⁡(1-z)+(-mz-log⁡(1-z))¯=z+∑n=2∞1nzn+(1-m)z¯+∑n=2∞1nz¯n(0<m≦1)
satisfies the condition of Theorem 2.1 with φ=0 and belongs to the class 𝒞ℋ. In particular, putting m=1, we obtain Figure 3.
By making use of Lemma1.8 with ɛ=0 and applying Lemma 1.9, we readily obtain the next theorem.Figure 3
The image off(z)=-z¯-2log|1-z|.Theorem 2.3.
Iff(z)∈ℋ is locally univalent in  𝕌 and satisfies
(2.8)∑n=2∞[|∑k=1n{∑j=1k(-1)k-jj(j+1)(αk-j)aj}(βn-k)|+|∑k=1n{∑j=1k(-1)k-jj(j-1)(αk-j)aj}(βn-k)|]≦2
for some real numbers α and β, then f(z)∈𝒞ℋ.Puttingα=β=0 in the above theorem, we arrive at the following result due to Jahangiri and Silverman [7].Theorem 2.4.
Iff(z)∈ℋ is locally univalent in  𝕌 with
(2.9)∑n=2∞n2|an|≦1,
then f(z)∈𝒞ℋ.Furthermore, takingα=1 and β=0 in the theorem, we have the following corollary.Corollary 2.5.
Iff(z)∈ℋ is locally univalent in  𝕌 and satisfies
(2.10)∑n=2∞{n|(n+1)an-(n-1)an-1|+(n-1)|nan-(n-2)an-1|}≦2,
then f(z)∈𝒞ℋ.Example 2.6.
The function(2.11)f(z)=-∫0zlog⁡(1-t)tdt+(z+(1-z)log⁡(1-z))¯=z+∑n=2∞1n2zn+∑n=2∞1n(n-1)z¯n
satisfies the conditions of Corollary 2.5 and belongs to the class 𝒞ℋ as shown in Figure 4.Figure 4
The image off(z)=-∫0z(log(1-t)/t)dt+(z+(1-z)log(1-z))¯.


## 3. Appendix

A sequence{cn}n=0∞ of nonnegative real numbers is called a convex null sequence if cn→0 as n→∞ and(3.1)cn-cn+1≧cn+1-cn+2≧0
for all n(n=0,1,2,…).The next lemma was obtained by Fejér [10].Lemma 3.1.
Let{cn}k=0∞ be a convex null sequence. Then, the function
(3.2)p(z)=c02+∑n=1∞cnzn
is analytic and satisfies Re(p(z))>0 in 𝕌.Applying the above lemma, we deduce the following theorem.Theorem 3.2.
For someb(|b|<1) and some convex null sequence  {cn}n=0∞ with  c0=2, the function
(3.3)f(z)=h(z)+g(z)¯=z+∑n=2∞cn-1nzn+b(z+∑n=2∞cn-1nzn)¯
belongs to the class 𝒞ℋ.Proof.
Let us defineF(z) by
(3.4)F(z)=h(z)+ɛg(z)1+ɛb=z+∑n=2∞cn-1nzn
for each ɛ(|ɛ|=1). Then, we know that
(3.5)F′(z)=c02+∑n=1∞cnzn(c0=2).
By virtue of Lemmas 1.7 and 3.1, it follows that Re(F′(z))>0(z∈𝕌), that is, F(z)∈𝒞. Thus, we conclude that f(z)=h(z)+g(z)¯∈𝒞ℋ.In the same manner, we also have the following theorem.Theorem 3.3.
For someb(|b|<1) and some convex null sequence  {cn}n=0∞ with  c0=2, the function
(3.6)f(z)=h(z)+g(z)¯=z+∑n=2∞1n(1+∑j=1n-1cj)zn+b(z+∑n=2∞1n(1+∑j=1n-1cj)zn)¯
belongs to the class 𝒞ℋ.Proof.
Let us defineF(z) by
(3.7)F(z)=h(z)+ɛg(z)1+ɛb=z+∑n=2∞1n(1+∑j=1n-1cj)zn
for each ɛ(|ɛ|=1). Then, we know that
(3.8)(1-z)F′(z)=c02+∑n=1∞cnzn(c0=2).
Therefore, by the help of Lemmas 1.7 and 3.1, we obtain that Re((1-z)F′(z))>0(z∈𝕌), that is, F(z)∈𝒞, which implies that f(z)=h(z)+g(z)¯∈𝒞ℋ.Remark 3.4.
The sequence(3.9){cn}n=0∞={2,1,23,…,2n+1,…}
is a convex null sequence because
(3.10)lim⁡n→∞cn=lim⁡n→∞(2n+1)=0,cn-cn+1=2(n+1)(n+2)≧0,(cn-cn+1)-(cn+1-cn+2)=4(n+1)(n+2)(n+3)≧0(n=0,1,2,…).
Settingb=1/4 in Theorem 3.2 with the above sequence {cn}n=0∞, we derive the following example.Example 3.5.
The function(3.11)f(z)=-z-2∫0zlog⁡(1-t)tdt-14(z+2∫0zlog⁡(1-t)tdt)¯=z+∑n=2∞2n2zn+14(z+∑n=2∞2n2zn)¯
is in the class 𝒞ℋ as shown in Figure 5.Figure 5
The image off(z) in Example 3.5.Moreover, we know the following remark.Remark 3.6.
The sequence(3.12){cn}n=0∞={2,1,12,…,21-n,…}
is a convex null sequence because
(3.13)lim⁡n→∞cn=lim⁡n→∞21-n=0,cn-cn+1=2-n≧0,(cn-cn+1)-(cn+1-cn+2)=2-(n+1)≧0(n=0,1,2,…).
Hence, lettingb=1/4 in Theorem 3.3 with the sequence {cn}n=0∞={21-n}n=0∞, we have the following example.Example 3.7.
The function(3.14)f(z)=-3log⁡(1-z)+4log⁡(1-z2)+(-34log⁡(1-z)+log⁡(1-z2))¯=z+∑n=2∞1n(1+∑j=1n-121-j)zn+14(z+∑n=2∞1n(1+∑j=1n-121-j)zn)¯
is in the class 𝒞ℋ as shown in Figure 6.Figure 6
The image off(z) in Example 3.7.



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