Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 576571, 9 pages
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, ranaab@hotmail.com
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.
Copyright q 2008 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 function f uiv 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 orientationpreserving in D is that |g z| < |h z| in D, see 1.
Denote by H 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 f0 h0 fz 0 − 1 0. Therefore, for f h g ∈ H, we can express h and g by the
following power series expansion:
hz z ∞
an zn ,
n2
gz ∞
bn zn ,
b1 < 1.
1.1
n1
Observe that H reduces S, the class of normalized univalent analytic functions, if the
coanalytic part of f is zero.
For f 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
Dn fz Dn hz Dn gz,
1.2
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International Journal of Mathematics and Mathematical Sciences
where the Ruscheweh derivative of a power series fz z Dn fz z
1 − zn1
n
n
n2 an z
is given by
∗f.
1.3
The operator ∗ stands for the Hadamard product or convolution of two power series
fz ∞
an zn ,
gz n1
∞
bn zn
1.4
n1
defined by
f∗gz ∞
an bn zn .
1.5
n1
In 3, Owa introduced the following definition.
Definition 1.1. Let the function fz 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
Dzλ fz :
d
1
Γ1 − λ dz
1
0
fζ
z − ζλ
dζ
0 ≤ λ < 1,
1.6
where the multiplicity of z − ζ−λ is removed by requiring logz − ζ to be real when z − ζ > 0.
In 4, Owa gave the relation between the fractional derivative and Ruscheweyh
n
operator for the function fz z ∞
n2 an z as
Dλ fz :
1
Dzλ zλ−1 fz ,
Γ1 λ
0 < λ < 1,
1.7
D0 fz lim Dλ fz,
λ→∞
D1 fz lim Dλ fz.
λ→1
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
Dzλ zλ−1 fz Dzλ zλ−1 hz Dzλ zλ−1 gz ,
Dz0 fz lim Dzλ fz,
λ→0
Dz1 fz lim Dzλ fz.
λ→1
0 < λ < 1,
1.8
R. A. Al-Khal and H. A. Al-Kharsani
3
Since Dλ f Dλ h Dλ g, it was proved in 1 that the harmonic function Dλ f is starlike of
order 1/2 if and only if the analytic function Dλ h − Dλ g is starlike of order 1/2, and it was
shown in 4, Theorem 3 that Dλ h − Dλ g is starlike of order 1/2 if and only if Re{Dzλ1 zλ h −
g/Dzλ zλ−1 h − g} > 1 λ/2 for 0 < λ < 1. Since Re{Dzλ1 zλ h − g/Dzλ zλ−1 h − g} ReΓ2 λDλ1 h − g/Γ1 λDλ h − g, then Dλ h − Dλ g is starlike of order 1 λΓ1 λ/2Γ2 λ, hence Dλ f Dλ h Dλ g is starlike of order 1 λΓ1 λ/2Γ2 λ. This means
Dzλ1 zλ f
DDλ f
1 λ
1 λΓ1 λ
Re λ >
⇒ Re λ .
>
λ−1
2Γ2
λ
2
D f
Dz z f
1.9
Recently, Owa and Srivastava 5 studied the linear Ωλ defined by operator
Ωλ fz : Γ2 − λzλ Dzλ fz
0 ≤ λ < 1,
1.10
where f is normalized and analytic function on U.
It is easily seen that
Ω1 f zf .
Ω0 f f,
1.11
Analogously, we studied the linear operator Ωλ defined on the harmonic function f h g
by
Ωλ fz Ωλ hz Ωλ gz,
1.12
where
Ωλ hz : Γ2 − λzλ Dzλ hz ∞
Γn 2Γ2 − λ
n0
Ωλ gz : Γ2 − λzλ Dzλ gz Γn 2 − λ
∞
Γn 2Γ2 − λ
n0
Γn 2 − λ
an1 zn1 ,
a1 1,
1.13
bn1 zn1 ,
b1 0.
We will define subclasses of normalized harmonic functions obtained by the Hadamard
product and using the fractional derivative.
2. Main results
Let h and g be analytic in U. Let PH stand for harmonic functions f h g so that Re f >
0, z ∈ U and f0 1.
If the function fz fz h g belongs to PH for the analytic and normalized functions
hz z ∞
an zn ,
gz n2
0
then the class of functions f h g is denoted by PH
6.
∞
n2
bn zn ,
2.1
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International Journal of Mathematics and Mathematical Sciences
The function
tα z z 1 2
1
z ··· zn · · ·
1α
1 n − 1α
2.2
is analytic on U when α is a complex number different from −1, −1/2, −1/3, . . . . For Ωλ f ∈
λ,0
0
, we denote by PH
α the class of functions defined by
PH
Ωλ F Ωλ f∗ tα tα .
2.3
Therefore,
Ωλ F Ωλ H Ωλ g
z
z
∞
∞
Γn 1Γ2 − λ
Γn 1Γ2 − λ
an zn bn zn
Γn
1
−
λ1
n
−
1α
Γn
1 − λ1 n − 1α
n2
n2
∞
n2
An zn ∞
Bn zn ,
2.4
z∈U
n2
λ,0
α. Conversely, if Ωλ F is in the form 2.4, with an , bn being the coefficients of Ωλ f ∈
is in PH
λ,0
0
, then Ωλ F PH
α.
PH
0,0
0,0
0
0
α ≡ PH
α 7 and PH
0 ≡ PH
.
Note that PH
λ,0
0
α, then there exists Ωλ f ∈ PH
so that
Theorem 2.1. If Ωλ F ∈ PH
α z Ωλ F z z zΩλ Fz z 1 − αΩλ Fz Ωλ fz.
2.5
λ,0
0
, there exists Ωλ F ∈ PH
α satisfying 2.5.
Conversely, for any function f such that Ωλ f ∈ PH
λ,0
0
α. If Ωλ f ∈ PH
, then since
Proof. Let Ωλ F ∈ PH
αztα z 1 − αtα z t0 z
2.6
as Ωλ F Ωλ f∗tα tα , we obtain that
Ωλ fz α Ωλ fz∗ ztα ztα 1 − α Ωλ fz∗ tα tα .
2.7
Ωλ fz α z Ωλ F z z z Ωλ F z z 1 − αΩλ Fz.
2.8
Therefore,
R. A. Al-Khal and H. A. Al-Kharsani
5
0
, from 2.1, 2.2, and 2.5,
Conversely, for Ωλ f ∈ PH
z
∞
Γn 2Γ2 − λ
n2
Γn 1 − λ
z
an zn ∞
Γn 1Γ2 − λ
Γn 1 − λ
n2
∞
1 n − 1αAn z n
n2
∞
bn zn
1 n − 1αBn
2.9
zn ,
n2
where
An Γn 1Γ2 − λ
an ,
Γn 1 − λ1 n − 1α
Bn Γn 1Γ2 − λ
bn .
Γn 1 − λ1 n − 1α
2.10
Therefore,
Ωλ F z ∞
∞
An zn Bn zn Ωλ f∗ tα z tα z .
n2
2.11
n2
λ,0
α, if and only if
Theorem 2.2. A function Ωλ F of the form 2.4 belongs to PH
Re z Ωλ Hz α Ωλ Gz Ωλ Hz Ωλ Gz
> 0,
z ∈ U.
2.12
0
α z Ωλ H z Ωλ G 1 − α Ωλ H Ωλ G Ωλ h Ωλ g ∈ PH
2.13
λ,0
α, then from Theorem 2.1,
Proof. If Ωλ F Ωλ H Ωλ G ∈ PH
and Ωλ h Ωλ g ∈ PH . Hence
λ λ Ω h Ω g
× Re αz Ωλ H α Ωλ H 1 − α Ωλ H αz Ωλ G α Ωλ G 1 − α Ωλ G
λ λ × Re z α Ωλ H α Ωλ G
Ω H Ω G .
2.14
0 < Re
Conversely, if the function Ωλ F Ωλ H Ωλ G of the form 2.4 satisfies 2.10, then by
Theorem 2.1 Ωλ h Ωλ g ∈ PH and the following function holds:
0
.
Ωλ f Ωλ h Ωλ g α z Ωλ H z Ωλ G 1 − α Ωλ H Ωλ G ∈ PH
λ,0
Then by Theorem 2.1, Ωλ F Ωλ H Ωλ G ∈ PH
α.
2.15
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International Journal of Mathematics and Mathematical Sciences
λ,0
α is convex and compact.
Theorem 2.3. PH
λ,0
Proof. Let Ωλ F1 Ωλ H1 Ωλ G1 , Ωλ F 2 Ωλ H2 Ωλ G2 ∈ PH
α and let μ ∈ 0, 1. Then
Re z α μ Ωλ H1 z 1 − μ Ωλ H2 z
α Ωλ G1 z 1 − μ Ωλ G2 z
λ
μ Ω H1 z Ωλ G1 z 1 − μ Ωλ H2 z Ωλ G2 z
λ
μ Re z α Ωλ H1 z α Ωλ G1 z
Ω H1 z Ωλ G1 z
λ
1 − μRe z α Ωλ H2 z α Ωλ G2 z
Ω H2 z Ωλ G2 z
2.16
> 0.
λ,0
λ,0
Hence from Theorem 2.2, μΩλ F 1 1 − μΩλ F2 ∈ PH
α. Therefore, PH
α is convex.
λ,0
λ
λ
λ
λ
λ
Now, let Ω Fn Ω Hn Ω Gn ∈ PH α and let Ω Fn → Ω F Ωλ H Ωλ G. By
Theorem 2.2,
0
.
α z Ωλ Hn z Ωλ Gn 1 − α Ωλ Hn Ωλ Gn ∈ PH
2.17
0
Since PH
is compact, see 6,
0
α z Ωλ H z Ωλ G 1 − α Ωλ H Ωλ G ∈ PH
.
2.18
λ,0
λ,0
Hence by Theorem 2.1, Ωλ F ∈ PH
α, therefore PH
α is compact.
λ,0
Theorem 2.4. If Ωλ F Ωλ H Ωλ G ∈ PH
α and |z| r < 1, then
−r 2 In1 r ≤ Re α z Ωλ Hn z Ωλ Gn 1 − α Ωλ Hn Ωλ Gn
≤ −r − 2 In1 − r.
2.19
Equality is obtained for the function 2.3 where
Ωλ f 2z ln1 − z − 3z − 3 ln1 − z,
z ∈ U.
2.20
λ,0
0
Proof. From Theorem 2.1, if Ωλ H Ωλ G ∈ PH
α, then there exists Ωλ f Ωλ h Ωλ g ∈ PH
so
that
α z Ωλ H zΩλ G 1 − α Ωλ H Ωλ G Ωλ f.
2.21
Since by 5, Proposition 2.2
−r 2 ln1 r ≤ Re Ωλ f ≤ −r − 2 ln1 − r,
this completes the proof.
2.22
R. A. Al-Khal and H. A. Al-Kharsani
7
λ,0
0
α and Re α > 0, then there exists Ωλ f ∈ PH
so that
Theorem 2.5. If Ωλ F Ωλ H Ωλ G ∈ PH
Ωλ F 1
α
1
ζ1/α−2 Ωλ f zζdζ,
z ∈ U.
2.23
0
Proof. Since
tα z 1
α
1
0
ζ1/α−1
z
dζ,
1 − zζ
|ζ| ≤ 1, Re α > 0,
2.24
0
and for Ωλ f Ωλ h Ωλ g ∈ PH
,
Ω h z ∗
λ
λ Ω h zζ
z
,
1 − zζ
ζ
λ Ω g zζ
z
Ω g z ∗
,
1 − zζ
ζ
λ
2.25
we have
1
Ωλ H z Ωλ h z ∗ tα α
1
Ω G z Ωλ g z ∗ tα α
λ
1
ζ1/α−2 Ωλ h zζdζ,
0
1
ζ
1/α−2
2.26
Ω g zζdζ.
λ
0
Hence Ωλ F is type 2.23.
0
Theorem 2.6. If Re α > 0, then P0λ,0 α ⊂ PH
.
λ,0
0
α and Re α > 0. Then there exists Ωλ f ∈ PH
so that
Proof. Let Ωλ F ∈ PH
Ω λ F Ω λ H Ω λ G Ω λ f ∗ t α tα Ω λ h ∗ t α Ω λ g ∗ tα .
2.27
Hence 0 < Re{Ωλ h Ωλ g } Re{Ωλ h Ωλ g } and since Re α > 0, Re{Ωλ H Ωλ G } > 0 and Ωλ H0 0, Ωλ H 0 1, Ωλ G0 0, Ωλ G 0 0. And hence Ωλ F ∈
0
.
PH
λ,0
Theorem 2.7. Let Ωλ F Ωλ H Ωλ G ∈ PH
α. Then
i
An | − |Bn ≤
2Γn 1Γ2 − λ
,
nΓn 1 − λ|1 n − 1α|
n ≥ 1,
2.28
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International Journal of Mathematics and Mathematical Sciences
ii if Ωλ F is sense-preserving, then
2n − 1
Γn 1Γ2 − λ
An ≤
,
n Γn 1 − λ|1 n − 1α|
n 1, 2, . . . ,
Γn 1Γ2 − λ
2n − 3
,
|Bn | ≤
n Γn 1 − λ|1 n − 1α|
n 2, 3, . . . .
2.29
Proof. By 2.10,
An | − |Bn Γn 1Γ2 − λ
an | − |bn .
Γn 1 − λ|1 n − 1α|
2.30
Also, by 6, Theorem 2.3, we have
an | − |bn ≤ 2 .
n
2.31
The required results are obtained.
On the other hand, from 2.10, it is known 6, Corollary 2.5 that
2n − 1
an ≤
,
n
2n − 3
bn ≤
.
n
2.32
Then we get the coefficient inequalities for P0λ,0 α.
Remark 2.8. Taking λ 0 in Theorems 2.1–2.7, we get the similar results in 7.
λ,0
Theorem 2.9. Let Ωλ F Ωλ H Ωλ G ∈ PH
α and sense-preserving in U, then for |z| r < 1,
λ αz Ω H 1 − αΩλ H ≤ 2r ln1 − r,
1−r
2
λ αz Ω G 1 − αΩλ G ≤ 3r − r 3 ln1 − r.
1−r
2.33
λ,0
Proof. From Theorems 2.1 and 2.2, if Ωλ F Ωλ H Ωλ G ∈ PH
α, then there exists Ωλ f 0
λ
λ
Ω h Ω g ∈ P such that
H
αz Ωλ H 1 − αΩλ H Ωλ h,
αz Ωλ G 1 − αΩλ G Ωλ g.
By 6, Theorem 3.5, we obtain the results.
2.34
R. A. Al-Khal and H. A. Al-Kharsani
9
Remark 2.10. Taking λ 0 and α 0 in Theorem 2.9, we get 6, Theorem 2.4.
3. Positive order
We say that the harmonic function f h g of the form 2.1 is in the class PH β, 0 ≤ β < 1
for |z| r if Re f > β and f0 1.
If the function fz 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 PH0 β.
Denote by PHλ,0 β, α the class of functions defined By 2.3 where Ωλ f ∈ PH0 β.
Many of our results can be rewritten for functions in the class PHλ,0 β, α. For instance,
see the following theorems.
λ,0
Theorem 3.1. If Ωλ F∈ P H β, α, then there exists Ωλ f ∈ PH0 β so that
α z Ωλ F z z z Ωλ F z z 1 − αΩλ Fz Ωλ fz.
3.1
Conversely, for any function f such that Ωλ f ∈ PH0 β, there exists Ωλ F ∈ PHλ,0 β, α satisfying
3.1.
Theorem 3.2. A function Ωλ F belongs to PHλ,0 β, α if and only if
Re z Ωλ Hz α Ωλ Gz Ωλ Hz Ωλ Gz
> β,
z ∈ U.
3.2
Theorem 3.3. If Ωλ F ∈ PHλ,0 β, α and Re α > 0, then there exists Ωλ f ∈ PH0 β so that
Ωλ F 1
α
1
ζ1/α−2 Ωλ f zζdζ,
z ∈ U.
3.3
0
Theorem 3.4. If Re α > 0, then PHλ,0 β, α ⊂ PH0 β.
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