On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 576571,9pages

doi:10.1155/2008/576571

Review Article

On Some Subclasses of Harmonic Functions

Defined by Fractional Calculus

R. A. Al-Khal and H. A. Al-Kharsani

Department of Mathematics, Faculty of Science, Girls College, P.O. Box 838, Dammam 31113, Saudi Arabia

Correspondence should be addressed to R. A. Al-Khal,[email protected]

Received 1 July 2008; Accepted 3 November 2008

Recommended by Teodor Bulboaca

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.

Copyrightq2008 R. A. Al-Khal and H. A. Al-Kharsani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A continuous functionf uivis a complex-valued harmonic function in a complex domain Cif bothuandvare real harmonic inC. In any simply connected domainD⊆C, we can write f hg, wherehandg are analytic inD. We callhthe analytic part andg the coanalytic part off. A necessary and sufficient condition forf to be locally univalent and orientation-preserving inDis that|gz|<|hz|inD, see1.

Denote by H the class of functions f hg which are harmonic univalent and orientation-preserving in the open unit diskU{z:|z|<1}so thatf hgis normalized byf0 h0 fz0−1 0. Therefore, forf hg ∈H, we can expresshandgby the

following power series expansion:

hz z

∞

n2

anzn, gz

∞

n1

bnzn, b1<1. 1.1

Observe that H reduces S, the class of normalized univalent analytic functions, if the coanalytic part offis zero.

For f hg given by1.1and n > −1, Murugusundaramoorthy2defined the Ruscheweyh derivative of the harmonic functionfhginHby

where the Ruscheweh derivative of a power seriesfz zn_n2anznis given by

Dnfz z

1−zn1∗f. 1.3

The operator∗stands for the Hadamard product or convolution of two power series

fz

∞

n1

anzn, gz

∞

n1

bnzn 1.4

defined by

f∗gz

∞

n1

anbnzn. 1.5

In3, Owa introduced the following definition.

Definition 1.1. Let the functionfzbe analytic in a simply connected domain of thez-plane containing the origin and let 0≤λ <1. The fractional derivative offof orderλis defined by

Dλ zfz:

1

Γ1−λ d dz

₁

0 fζ

z−ζλdζ 0≤λ <1, 1.6

where the multiplicity ofz−ζ−λis removed by requiring logz−ζto be real whenz−ζ >0.

In 4, Owa gave the relation between the fractional derivative and Ruscheweyh operator for the functionfz z∞_n2anznas

Dλfz: 1

Γ1λD λ z

zλ−1fz, 0< λ <1,

D0fz lim

λ→ ∞D

λ_fz,

D1fz lim

λ→1D λ_fz.

1.7

Using 1.2 and the relation between the fractional derivative and Ruscheweyh operator, we define the fractional derivative of orderλ, 0≤λ < 1, for the harmonic function fhgas

Dλ_zzλ−1fzDλ_zzλ−1hzDλz

zλ−1_{gz, 0< λ <1,}

D0_zfz lim

λ→0D λ zfz,

D1_zfz lim

λ→1D λ zfz.

SinceDλ_{f D}λ_hDλ_{g,it was proved in1that the harmonic functionD}λ_{f is starlike of}

order 1/2 if and only if the analytic functionDλ_h−Dλ_{g is starlike of order 1/2, and it was}

shown in4, Theorem 3thatDλ_h−Dλ_{gis starlike of order 1/2 if and only if Re{D}λ1 z zλh− g/Dλ

zzλ−1h−g}>1λ/2 for 0< λ <1. Since Re{Dzλ1zλh−g/Dzλzλ−1h−g}

ReΓ2λDλ1h−g/Γ1λDλ_{h−g, thenD}λ_h−Dλ_{gis starlike of order1λΓ1} λ/2Γ2λ, henceDλ_fDλ_hDλ_{gis starlike of order1λΓ1λ/2Γ2λ. This means}

ReDD

λ_f

Dλ_f >

1λΓ1λ

2Γ2λ ⇒Re

D_zλ1zλf Dzλ

zλ−1_f > 1λ

2 . 1.9

Recently, Owa and Srivastava5studied the linearΩλ_{defined by operator}

Ωλ_{fz: Γ2−λz}λ_Dλ

zfz 0≤λ <1, 1.10

wherefis normalized and analytic function onU. It is easily seen that

Ω0_{ff, Ω}1_{fzf. 1.11}

Analogously, we studied the linear operatorΩλ_{defined on the harmonic functionf hg}

by

Ωλ_{fz Ω}λ_{hz Ω}λ_{gz, 1.12}

where

Ωλ_{hz: Γ2−λz}λ_Dλ zhz

∞

n0

Γn2Γ2−λ

Γn2−λ an1z n1_{, a}

11,

Ωλ_{gz: Γ2−λz}λ_Dλ zgz

∞

n0

Γn2Γ2−λ

Γn2−λ bn1z n1_{, b}

1 0.

1.13

We will define subclasses of normalized harmonic functions obtained by the Hadamard product and using the fractional derivative.

2. Main results

Lethandg be analytic inU. LetPHstand for harmonic functionsf hgso that Ref >

0, z∈Uandf0 1.

If the functionfzfzhgbelongs toPHfor the analytic and normalized functions

hz z

∞

n2

anzn, gz

∞

n2

bnzn, 2.1

The function

tαz z 1

1αz

2_{· · ·} 1

1 n−1αz

n_{· · · 2.2}

is analytic onUwhenαis a complex number different from−1,−1/2,−1/3, . . . .ForΩλ_f∈

P0

H, we denote byP λ,0

H αthe class of functions defined by

Ωλ_{F Ω}λ_{f∗ t} αtα

. 2.3

Therefore,

Ωλ_{F Ω}λ_{H Ω}λ_g

z

∞

n2

Γn1Γ2−λ

Γn1−λ1 n−1αanz n∞

n2

Γn1Γ2−λ

Γn1−λ1 n−1αbnz n

z

∞

n2 Anzn

∞

n2

Bnzn, z∈U

2.4

is inP_Hλ,0α. Conversely, ifΩλ_{Fis in the form2.4, witha}

n, bnbeing the coefficients ofΩλf∈

P0

H, thenΩλFP λ,0 H α.

Note thatP_H0,0α≡P_H0α 7andP_H0,00≡P_H0.

Theorem 2.1. IfΩλ_F∈Pλ,0

H α, then there existsΩλf ∈PH0 so that

αz ΩλF_zz zΩλF_zz 1−αΩλFz Ωλfz. 2.5

Conversely, for any functionfsuch thatΩλ_{f ∈P}0

H, there existsΩλF∈P λ,0

H αsatisfying2.5.

Proof. LetΩλ_{F ∈P}λ,0

H α. IfΩλf∈PH0, then since

αzt_αz 1−αtαz t0z 2.6

asΩλ_{F Ω}λ_f∗t

αtα, we obtain that

Ωλ_{fz αΩ}λ_{fz∗ zt}

αztα

1−αΩλfz∗ tαtα

. 2.7

Therefore,

Ωλ_{fz αz Ω}λ_F

zz z Ω λ_F

zz

Conversely, forΩλ_{f ∈P}0

H, from2.1,2.2, and2.5,

z

∞

n2

Γn2Γ2−λ

Γn1−λ anz n∞

n2

Γn1Γ2−λ

Γn1−λ bnz n

z

∞

n2

1 n−1αAnzn

∞

n2

1 n−1αBnzn,

2.9

where

An _Γ Γn1Γ2−λ

n1−λ1 n−1αan, Bn

Γn1Γ2−λ

Γn1−λ1 n−1αbn. 2.10

Therefore,

Ωλ_Fz∞ n2

Anzn

∞

n2

Bnzn Ωλf∗

tαz tαz

. 2.11

Theorem 2.2. A functionΩλ_{Fof the form2.4belongs toP}λ,0

H α, if and only if

Rez Ωλ_{Hzα Ω}λ_{Gz Ω}λ_{Hz Ω}λ_{Gz>0, z∈U. 2.12}

Proof. IfΩλ_{F Ω}λ_{H Ω}λ_G∈Pλ,0

H α, then fromTheorem 2.1,

αz ΩλHz Ωλ_{G 1−αΩ}λ_{H Ω}λ_{G Ω}λ_{h Ω}λ_{g ∈P}0

H 2.13

andΩλ_{h Ω}λ_g∈P

H. Hence

0<Re Ωλh Ωλ_g

×Reαz ΩλHα ΩλH 1−α ΩλHαz ΩλGα ΩλG 1−α ΩλG

×Rez α ΩλHα ΩλG ΩλH ΩλG.

2.14

Conversely, if the function Ωλ_{F Ω}λ_{H Ω}λ_{G of the form 2.4 satisfies 2.10, then by}

Theorem 2.1Ωλ_{h Ω}λ_g∈P

Hand the following function holds:

Ωλ_{f Ω}λ_{h Ω}λ_{gαz Ω}λ_{Hz Ω}λ_{G 1−αΩ}λ_{H Ω}λ_G∈P0

H. 2.15

Theorem 2.3. P_Hλ,0αis convex and compact.

Proof. LetΩλ_F

1 ΩλH1 ΩλG1, ΩλF2 ΩλH2 ΩλG2∈P_Hλ,0αand letμ∈0,1. Then

Rezα μ ΩλH1z

1−μ ΩλH2z

α ΩλG1z

1−μ ΩλG2z

μ ΩλH1z

Ωλ_G 1z

1−μ ΩλH2z

Ωλ_G 2z

μRezα ΩλH1z

α ΩλG1z

_Ωλ_H 1z

_Ωλ_G 1z

1−μRezα ΩλH2z

_{α Ω}λ_G 2z

_Ωλ_H 2z

_Ωλ_G 2z

>0.

2.16

Hence fromTheorem 2.2,μΩλ_F

1 1−μΩλF2∈P_Hλ,0α. Therefore,P_Hλ,0αis convex.

Now, letΩλFn ΩλHn ΩλGn ∈ P_Hλ,0αand letΩλFn → ΩλF ΩλH ΩλG. By

Theorem 2.2,

αz ΩλHn

z Ωλ_{Gn 1−αΩ}λ_H

n ΩλGn

∈P_H0. 2.17

SinceP0

His compact, see6,

αz ΩλHz Ωλ_{G 1−αΩ}λ_{H Ω}λ_G∈P0

H. 2.18

Hence byTheorem 2.1,Ωλ_F∈Pλ,0

H α, thereforeP λ,0

H αis compact.

Theorem 2.4. IfΩλ_{F Ω}λ_{H Ω}λ_G∈Pλ,0

H αand|z|r <1, then

−r2 In1r≤Reαz ΩλHn

z Ωλ_{Gn 1−αΩ}λ_H

n ΩλGn

≤ −r−2 In1−r.

2.19

Equality is obtained for the function2.3where

Ωλ_{f 2zln1−z−3z−3 ln1−z, z∈U. 2.20}

Proof. FromTheorem 2.1, ifΩλ_{H Ω}λ_G∈Pλ,0

H α, then there existsΩλf Ωλh Ωλg∈PH0 so

that

αz ΩλHzΩλ_{G 1−αΩ}λ_{H Ω}λ_{G Ω}λ_{f. 2.21}

Since by5, Proposition 2.2

−r2 ln1r≤Re Ωλf≤ −r−2 ln1−r, 2.22

Theorem 2.5. IfΩλ_{F Ω}λ_{H Ω}λ_G∈Pλ,0

H αand Reα >0, then there existsΩλf∈PH0 so that

Ωλ_F 1 α

₁

0

ζ1/α−2 Ωλfzζdζ, z∈U. 2.23

Proof. Since

tαz

1 α

1

0

ζ1/α−1 z

1−zζdζ, |ζ| ≤1, Reα >0, 2.24

and forΩλ_{f Ω}λ_{h Ω}λ_g∈_P0 H,

Ωλ_hz∗ z

1−zζ

Ωλ_hzζ

ζ , Ω

λ_gz∗ z

1−zζ

Ωλ_gzζ

ζ , 2.25

we have

Ωλ_{Hz Ω}λ_hz∗t α 1

α

1

0

ζ1/α−2 Ωλhzζdζ,

Ωλ_{Gz Ω}λ_gz∗t α

1 α

₁

0

ζ1/α−2 Ωλgzζdζ.

2.26

HenceΩλ_{Fis type2.23.}

Theorem 2.6. If Reα >0, thenP₀λ,0α⊂P0 H.

Proof. LetΩλ_{F ∈P}λ,0

H αand Reα >0. Then there existsΩλf∈PH0 so that

Ωλ_{F Ω}λ_{H Ω}λ_{G Ω}λ_{f∗ t} αtα

Ωλ_h∗t α

Ωλ_g∗t α

. 2.27

Hence 0 < Re{Ωλ_{h Ω}λ_{g} Re{Ω}λ_{h Ω}λ_{g} and since Reα > 0, Re{Ω}λ_H Ωλ_{G}>0 andΩ}λ_{H0 0, Ω}λ_{H0 1, Ω}λ_{G0 0, Ω}λ_{G0 0. And henceΩ}λ_{F ∈}

P0 H.

Theorem 2.7. LetΩλ_{F Ω}λ_{H Ω}λ_G∈Pλ,0

H α. Then i

An| − |Bn≤

2Γn1Γ2−λ

iiifΩλ_{Fis sense-preserving, then}

An≤

2n−1 n

Γn1Γ2−λ

Γn1−λ|1 n−1α|, n1,2, . . . ,

|Bn| ≤ 2n−3 n

Γn1Γ2−λ

Γn1−λ|1 n−1α|, n2,3, . . . .

2.29

Proof. By2.10,

_{An| − |Bn} Γn1Γ2−λ

Γn1−λ|1 n−1α|an| − |bn. 2.30

Also, by6, Theorem 2.3, we have

_{an| − |bn≤} 2

n. 2.31

The required results are obtained.

On the other hand, from2.10, it is known6, Corollary 2.5that

_an≤ 2n−1

n , bn≤

2n−3

n . 2.32

Then we get the coefficient inequalities forP₀λ,0α.

Remark 2.8. Takingλ0 in Theorems2.1–2.7, we get the similar results in7.

Theorem 2.9. LetΩλ_{F Ω}λ_{H Ω}λ_G∈Pλ,0

H αand sense-preserving inU, then for|z|r <1,

_{αz Ω}λ_{H 1−αΩ}λ_H≤ 2r

1−r ln1−r,

αz ΩλG 1−αΩλG≤ 3r−r 2

1−r 3 ln1−r.

2.33

Proof. From Theorems2.1and2.2, ifΩλ_{F Ω}λ_{H Ω}λ_{G ∈ P}λ,0

H α, then there existsΩλf Ωλ_{h Ω}λ_g∈P0

Hsuch that

αz ΩλH 1−αΩλH Ωλh,

αz ΩλG 1−αΩλG Ωλg.

2.34

Remark 2.10. Takingλ0 andα0 inTheorem 2.9, we get6, Theorem 2.4.

3. Positive order

We say that the harmonic functionf hgof the form2.1is in the classPHβ, 0≤β <1

for|z|rif Ref > βandf0 1.

If the functionfz fz hg belongs to PHβ for the analytic and normalized

functionshandgof the form2.1, then the class of functionsfhgis denoted byP_H0β. Denote byP_Hλ,0β, αthe class of functions defined By2.3whereΩλ_f∈P0

Hβ.

Many of our results can be rewritten for functions in the classP_Hλ,0β, α. For instance, see the following theorems.

Theorem 3.1. IfΩλ_F∈Pλ,0

H β, α, then there existsΩλf ∈PH0βso that

αz ΩλF_zz z ΩλF_zz 1−αΩλFz Ωλfz. 3.1

Conversely, for any function f such thatΩλ_{f ∈ P}0

Hβ, there existsΩλF ∈ P λ,0

H β, α satisfying 3.1.

Theorem 3.2. A functionΩλ_{Fbelongs toP}λ,0

H β, αif and only if

Rez ΩλHzα ΩλGz ΩλHz ΩλGz> β, z∈U. 3.2

Theorem 3.3. IfΩλ_F∈Pλ,0

H β, αand Re α >0, then there existsΩλf∈PH0βso that

Ωλ_F 1 α

₁

0

ζ1/α−2 Ωλfzζdζ, z∈U. 3.3

Theorem 3.4. If Reα >0, thenP_Hλ,0β, α⊂P0 Hβ.

References

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