# On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

Authors: R. A. Al-Khal; H. A. Al-Kharsani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

The purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.


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## Body

## 1. Introduction

A continuous
functionf=u+iv is a complex-valued  harmonic function in a
complex domain C if both u and v are real harmonic in C.
In any simply connected domain D⊆C,
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 D is that |g′(z)|<|h′(z)| in D,
see [1].Denote byH the class of functions f=h+g¯ which are harmonic univalent and
orientation-preserving in the open unit disk U={z:|z|<1} so that f=h+g¯ is normalized by f(0)=h(0)=fz(0)−1=0.
Therefore, for f=h+g¯∈H,
we can express h and g by the following power series expansion:(1.1)h(z)=z+∑n=2∞anzn,g(z)=∑n=1∞bnzn,|b1|<1.Observe that H reduces S,
the class of normalized univalent analytic functions, if the coanalytic part of f is zero.Forf=h+g¯ given by (1.1) and n>−1,
Murugusundaramoorthy [2] defined the Ruscheweyh derivative of the harmonic
function f=h+g¯ in H by(1.2)Dnf(z)=Dnh(z)+Dng(z)¯,where the Ruscheweh derivative
of a power series f(z)=z+∑n=2nanzn is given by (1.3)Dnf(z)=z(1−z)n+1*f. The operator * stands for the Hadamard product or convolution of two power series(1.4)f(z)=∑n=1∞anzn,g(z)=∑n=1∞bnzndefined
by(1.5)(f*g)(z)=∑n=1∞anbnzn.In [3], Owa introduced the
following definition.Definition 1.1.
Let the functionf(z) be analytic in a simply connected domain of
the z-plane containing the origin and let 0≤λ<1.
The fractional derivative of f of order λ is defined by(1.6)Dzλf(z):=1Γ(1−λ)ddz∫01f(ζ)(z−ζ)λdζ(0≤λ<1),where the multiplicity of (z−ζ)−λ is removed by requiring log(z−ζ) to be real when z−ζ>0.In [4], Owa gave the relation between the fractional
derivative and Ruscheweyh operator for the function f(z)=z+∑n=2∞anzn as(1.7)Dλf(z):=1Γ(1+λ)Dzλ[zλ−1f(z)],0<λ<1,D0f(z)=limλ→∞Dλf(z),D1f(z)=limλ→1Dλf(z).Using (1.2) and the relation between the fractional
derivative and Ruscheweyh operator, we define the fractional derivative of
order λ, 0≤λ<1,
for the harmonic function f=h+g¯ as(1.8)Dzλ[zλ−1f(z)]=Dzλ[zλ−1h(z)]+Dzλ[zλ−1g(z)]¯,0<λ<1,Dz0f(z)=limλ→0Dzλf(z),Dz1f(z)=limλ→1Dzλf(z).Since Dλf=Dλh+Dλg¯,
it was proved in [1]
that the harmonic function Dλf  is starlike of order 1/2 if and only if the analytic function Dλh−Dλg is starlike of order 1/2, and it was shown in
[4, Theorem 3] that Dλh−Dλg is starlike of order 1/2 if and only if Re{Dzλ+1[zλ(h−g)]/Dzλ[zλ−1h−g)]}>(1+λ)/2  for 0<λ<1.
Since Re{Dzλ+1[zλ(h−g)]/Dzλ[zλ−1(h−g)]}=Re(Γ(2+λ)Dλ+1(h−g)/Γ(1+λ)Dλ(h−g)),
then Dλh−Dλg is starlike of order (1+λ)Γ(1+λ)/2Γ(2+λ),
hence Dλf=Dλh+Dλg¯ is starlike of order (1+λ)Γ(1+λ)/2Γ(2+λ). This means(1.9)ReDDλfDλf>(1+λ)Γ(1+λ)2Γ(2+λ)⇒ReDzλ+1[zλf]Dzλ[zλ−1f]>(1+λ)2.Recently, Owa and Srivastava [5] studied the linear Ωλ defined  by operator(1.10)Ωλf(z):=Γ(2−λ)zλDzλf(z)(0≤λ<1),where f is normalized and analytic function on U.It is easily seen that(1.11)Ω0f=f,Ω1f=zf′.Analogously, we studied the
linear operator Ωλ defined on the harmonic function f=h+g¯ by(1.12)Ωλf(z)=Ωλh(z)+Ωλg(z)¯,where(1.13)Ωλh(z):=Γ(2−λ)zλDzλh(z)=∑n=0∞Γ(n+2)Γ(2−λ)Γ(n+2−λ)an+1zn+1,a1=1,Ωλg(z):=Γ(2−λ)zλDzλg(z)=∑n=0∞Γ(n+2)Γ(2−λ)Γ(n+2−λ)bn+1zn+1,b1=0.We will define subclasses of
normalized harmonic functions obtained by the Hadamard product and using the
fractional derivative.


## 2. Main Results

Leth and g be analytic in U.
Let PH stand for harmonic functions f=h+g¯ so that Ref>0,z∈U and f(0)=1.If the functionfz+fz¯=h′+g′¯ belongs to PH for the analytic and normalized functions(2.1)h(z)=z+∑n=2∞anzn,g(z)=∑n=2∞bnzn,then the class of functions f=h+g¯ is denoted by P˜H0 [6].The function(2.2)tα(z)=z+11+αz2+⋯+11+(n−1)αzn+⋯is analytic on U when α is a complex number different from −1,−(1/2),−(1/3),….
For Ωλf∈P˜H0,
we denote by P˜Hλ,0(α) the class of functions defined by(2.3)ΩλF=Ωλf*(tα+tα¯).Therefore,(2.4)ΩλF=ΩλH+Ωλg¯=z+∑n=2∞Γ(n+1)Γ(2−λ)Γ(n+1−λ)[1+(n−1)α]anzn+∑n=2∞Γ(n+1)Γ(2−λ)Γ(n+1−λ)[1+(n−1)α]bnzn¯=z+∑n=2∞Anzn+∑n=2∞Bnzn¯,z∈Uis in P˜Hλ,0(α).
Conversely, if ΩλF is in the form (2.4), with an,bn being the coefficients of Ωλf∈P˜H0,
then ΩλF=P˜Hλ,0(α).Note thatP˜H0,0(α)≡P˜H0(α) [7] and P˜H0,0(0)≡P˜H0.Theorem 2.1.
IfΩλF∈P˜Hλ,0(α),
then there exists Ωλf∈P˜H0 so that(2.5)α[z(ΩλF)z(z)+z¯(ΩλF)z¯(z)]+(1−α)ΩλF(z)=Ωλf(z).Conversely, for any function f such that Ωλf∈P˜H0,
there exists ΩλF∈P˜Hλ,0(α) satisfying (2.5).Proof.
LetΩλF∈P˜Hλ,0(α). If Ωλf∈P˜H0,
then since(2.6)αztα′(z)+(1−α)tα(z)=t0(z)as ΩλF=Ωλf*(tα+tα¯),
we obtain that(2.7)Ωλf(z)=α[Ωλf(z)*(ztα′+ztα′¯)]+(1−α)[Ωλf(z)*(tα+tα¯)].Therefore,(2.8)Ωλf(z)=α[z(ΩλF)z(z)+z¯(ΩλF)z¯(z)]+(1−α)ΩλF(z).Conversely, for Ωλf∈P˜H0,
from (2.1), (2.2), and (2.5),(2.9)z+∑n=2∞Γ(n+2)Γ(2−λ)Γ(n+1−λ)anzn+∑n=2∞Γ(n+1)Γ(2−λ)Γ(n+1−λ)bnzn¯=z+∑n=2∞[1+(n−1)α]Anzn+∑n=2∞[1+(n−1)α]Bnzn¯,where(2.10)An=Γ(n+1)Γ(2−λ)Γ(n+1−λ)[1+(n−1)α]an,Bn=Γ(n+1)Γ(2−λ)Γ(n+1−λ)[1+(n−1)α]bn.Therefore,(2.11)ΩλF=z+∑n=2∞Anzn+∑n=2∞Bnzn¯=Ωλf*[tα(z)+tα¯(z)].Theorem 2.2.
A functionΩλF of the form (2.4) belongs to P˜Hλ,0(α),
if and only if(2.12)Re{z(ΩλH(z))′′+α¯(ΩλG(z))′′+(ΩλH(z))′+(ΩλG(z))′}>0,z∈U.Proof.
IfΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α),
then  from Theorem 2.1,(2.13)α[z(ΩλH)′+z(ΩλG)′¯]+(1−α)[ΩλH+ΩλG¯]=Ωλh+Ωλg¯∈P˜H0and (Ωλh)′+(Ωλg)′¯∈PH.
Hence(2.14)0<Re{(Ωλh)′+(Ωλg)′¯}×Re{αz(ΩλH)′′+α(ΩλH)′+(1−α)(ΩλH)′+α¯z(ΩλG)′′+α¯(ΩλG)′+(1−α¯)(ΩλG)′}×Re{z(α(ΩλH)′′+α¯(ΩλG)′′)+(ΩλH)′+(ΩλG)′}.Conversely, if  the  function ΩλF=ΩλH+ΩλG¯ of the form (2.4) satisfies (2.10), then by
Theorem 2.1  (Ωλh)′+(Ωλg)′¯∈PH and the following function
holds:(2.15)Ωλf=Ωλh+Ωλg¯=α[z(ΩλH)′+z(ΩλG)′¯]+(1−α)[ΩλH+ΩλG¯]∈P˜H0.Then by Theorem 2.1, ΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α).Theorem 2.3.
P˜Hλ,0(α) is convex and compact.Proof.
LetΩλF1=ΩλH1+ΩλG1¯,ΩλF2=ΩλH2+ΩλG2¯∈P˜Hλ,0(α) and let μ∈[0,1].
Then(2.16)Re{z[α(μ(ΩλH1(z))′′+(1−μ)(ΩλH2(z))′′)+α¯(ΩλG1(z))′′+(1−μ)(ΩλG2(z))′′)]+μ[(ΩλH1(z))′+(ΩλG1(z))′]+(1−μ)[(ΩλH2(z))′+(ΩλG2(z))′]}=μRe{z[α(ΩλH1(z))′′+α¯(ΩλG1(z))′′]+(ΩλH1(z))′+(ΩλG1(z))′}+(1−μ)Re{z[α(ΩλH2(z))′′+α¯(ΩλG2(z))′′]+(ΩλH2(z))′+(ΩλG2(z))′}>0.Hence from Theorem 2.2, μΩλF1+(1−μ)ΩλF2∈P˜Hλ,0(α).
Therefore, P˜Hλ,0(α) is convex.
Now, letΩλFn=ΩλHn+ΩλGn¯∈P˜Hλ,0(α) and let ΩλFn→ΩλF=ΩλH+ΩλG¯.
By Theorem 2.2,(2.17)α[z(ΩλHn)′+z(ΩλGn)′¯]+(1−α)[ΩλHn+ΩλGn¯]∈P˜H0.Since P˜H0 is compact, see [6],(2.18)α[z(ΩλH)′+z(ΩλG)′¯]+(1−α)[ΩλH+ΩλG¯]∈P˜H0.Hence by Theorem 2.1, ΩλF∈P˜Hλ,0(α),
therefore P˜Hλ,0(α) is compact.Theorem 2.4.
IfΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α) and |z|=r<1,
then(2.19)−r+2In(1+r)≤Re{α[z(ΩλHn)′+z(ΩλGn)′¯]+(1−α)[ΩλHn+ΩλGn¯]}≤−r−2In(1−r).Equality is obtained for the
function (2.3) where(2.20)Ωλf=2z+ln(1−z)−3z¯−3ln(1−z¯),z∈U.Proof.
From Theorem2.1, if ΩλH+ΩλG¯∈P˜Hλ,0(α), then there exists Ωλf=Ωλh+Ωλg¯∈P˜H0 so that(2.21)α[z(ΩλH)′+z(ΩλG)′¯]+(1−α)[ΩλH+ΩλG¯]=Ωλf.Since by
[5, Proposition 2.2](2.22)−r+2ln(1+r)≤Re(Ωλf)≤−r−2ln(1−r),this completes the proof.Theorem 2.5.
IfΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α) and Reα>0,
then there exists Ωλf∈P˜H0  so that(2.23)ΩλF=1α∫01ζ1/α−2(Ωλf)(zζ)dζ,z∈U.Proof.
Since(2.24)tα(z)=1α∫01ζ1/α−1z1−zζdζ,|ζ|≤1,Reα>0,and for Ωλf=Ωλh+Ωλg¯∈P˜H0,(2.25)(Ωλh)(z)*z1−zζ=(Ωλh)(zζ)ζ,(Ωλg)(z)*z1−zζ=(Ωλg)(zζ)ζ,we have(2.26)(ΩλH)(z)=(Ωλh)(z)*tα=1α∫01ζ1/α−2(Ωλh)(zζ)dζ,(ΩλG)(z)=(Ωλg)(z)*tα=1α∫01ζ1/α−2(Ωλg)(zζ)dζ.Hence ΩλF is type (2.23).Theorem 2.6.
IfReα>0,
then P˜0λ,0(α)⊂P˜H0.Proof.
LetΩλF∈P˜Hλ,0(α) and Reα>0.
Then there exists Ωλf∈P˜H0 so that(2.27)ΩλF=ΩλH+ΩλG¯=Ωλf*(tα+tα¯)=(Ωλh*tα)+(Ωλg*tα¯).Hence 0<Re{(Ωλh)′+(Ωλg)′¯}=Re{(Ωλh)′+(Ωλg)′} and since Reα>0, Re{(ΩλH)′+(ΩλG)′}>0 and ΩλH(0)=0,(ΩλH)′(0)=1,ΩλG(0)=0,(ΩλG)′(0)=0.
And hence ΩλF∈P˜H0.Theorem 2.7.
LetΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α).
Then
(i)(2.28)∥An|−|Bn∥≤2Γ(n+1)Γ(2−λ)nΓ(n+1−λ)|(1+(n−1)α|,n≥1,
(ii) ifΩλF is sense-preserving, then(2.29)|An|≤2n−1nΓ(n+1)Γ(2−λ)Γ(n+1−λ)|1+(n−1)α|,n=1,2,…,|Bn|≤2n−3nΓ(n+1)Γ(2−λ)Γ(n+1−λ)|1+(n−1)α|,n=2,3,….Proof.
By (2.10),(2.30)∥An|−|Bn∥=Γ(n+1)Γ(2−λ)Γ(n+1−λ)|1+(n−1)α|∥an|−|bn∥.Also, by
[6, Theorem 2.3], we
have(2.31)∥an|−|bn∥≤2n.The required results are
obtained.
On the other hand, from (2.10), it is known
[6, Corollary 2.5] that(2.32)|an|≤2n−1n,|bn|≤2n−3n.Then we get the coefficient
inequalities for P˜0λ,0(α).Remark 2.8.
Takingλ=0 in Theorems 2.1–2.7, we get the similar
results in [7].Theorem 2.9.
LetΩλF=ΩλH=ΩλG¯∈P˜Hλ,0(α) and sense-preserving in U,
then for |z|=r<1,(2.33)|αz(ΩλH)′+(1−α)ΩλH|≤2r1−r+ln(1−r),|α¯z(ΩλG)′+(1−α¯)ΩλG|≤3r−r21−r+3ln(1−r).Proof.
From Theorems2.1 and 2.2, if ΩλF=ΩλH+ΩλG¯∈P˜Hλ,0(α),
then there exists Ωλf=Ωλh+Ωλg¯∈P˜H0 such that(2.34)αz(ΩλH)′+(1−α)ΩλH=Ωλh,α¯z(ΩλG)′+(1−α¯)ΩλG=Ωλg.By [6, Theorem 3.5], we obtain
the results.Remark 2.10.
Takingλ=0 and α=0 in Theorem 2.9, we get
[6, Theorem 2.4].


## 3. Positive Order

We say that
the harmonic functionf=h+g¯ of the form (2.1) is in the class PH(β),0≤β<1 for |z|=r if Ref>β and f(0)=1.If the functionfz+f¯z¯=h′+g′¯ belongs to PH(β) for the analytic and normalized functions h and g of the form (2.1), then the class of functions f=h+g¯ is denoted by P^H0(β).Denote byP^Hλ,0(β,α) the  class of functions defined By (2.3) where Ωλf∈P^H0(β).Many of our results can be rewritten for functions in
the classP^Hλ,0(β,α).
For instance, see the following theorems.Theorem 3.1.
IfΩλF∈P^Hλ,0(β,α),
then there exists Ωλf∈P^H0(β) so that(3.1)α[z(ΩλF)z(z)+z¯(ΩλF)z¯(z)]+(1−α)ΩλF(z)=Ωλf(z).Conversely, for any function f such that Ωλf∈P^H0(β),
there exists ΩλF∈P^Hλ,0(β,α) satisfying (3.1).Theorem 3.2.
A functionΩλF belongs to P^Hλ,0(β,α) if and only if(3.2)Re{z(ΩλH(z))′′+α¯(ΩλG(z))′′+(ΩλH(z))′+(ΩλG(z))′}>β,z∈U.Theorem 3.3.
IfΩλF∈P^Hλ,0(β,α) and Reα>0,
then there exists Ωλf∈P^H0(β) so that(3.3)ΩλF=1α∫01ζ1/α−2(Ωλf)(zζ)dζ,z∈U.Theorem 3.4.
IfReα>0,
then P^Hλ,0(β,α)⊂P^H0(β).



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