¨ Ozlem G ¨uney Department of Mathematics, Faculty of Science and Letters, Dicle University, 21280 Diyarbakir, Turkey Correspondence should be addressed to Sevtap S ¨umer Eker, sevtaps@d
Trang 1Volume 2008, Article ID 452057, 10 pages
doi:10.1155/2008/452057
Research Article
A New Subclass of Analytic Functions Involving
Al-Oboudi Differential Operator
Sevtap S ¨umer Eker and H ¨ Ozlem G ¨uney
Department of Mathematics, Faculty of Science and Letters, Dicle University, 21280 Diyarbakir, Turkey
Correspondence should be addressed to Sevtap S ¨umer Eker, sevtaps@dicle.edu.tr
Received 25 September 2007; Accepted 4 February 2008
Recommended by Jozsef Szabados
The main object of this paper is to introduce and investigate a new subclass of normalized analytic functions in the open unit disc U which is defined by Al-Oboudi differential operator Coefficient inequalities, extreme points, and integral means inequalities for fractional derivative for this class are given.
Copyright q 2008 S S ¨umer Eker and H ¨Ozlem G ¨uney 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 and definitions
LetA denote the class of functions f normalized by
f z z ∞
j2
which are analytic in the open unit discU {z : |z| < 1}.
For f ∈ A, Al-Oboudi 1 introduced the following operator:
D1f z 1 − δfz δzfz D δ f z, δ ≥ 0 1.3
D n f z D δ
D n−1f z, n ∈ N 1, 2, 3, . 1.4
If f is given by1.1, then from 1.3 and 1.4 we see that
D n f z z ∞
j2
1 j − 1δ n a j z j ,
When δ 1, we get Sˇalˇagean differential operator 2
Trang 2Definition 1.1 LetSm,n,δ α denote the subclass of A consisting of functions f which satisfy the
inequality
Re
D m f z
D n f z
for some 0≤ α < 1, m ∈ N, n ∈ N0, and all z∈ U
The object of the present paper is to investigate the coefficient bounds, extreme points, and integral mean inequalities for fractional derivatives of functions belonging to the class
Sm,n,δ α.
2 Coefficient inequalities
Our first theorem gives a sufficient condition for f ∈ A to belong to the class Sm,n,δ α.
Theorem 2.1 Let fz ∈ A satisfy
∞
j2
where
Ψm, n, j, δ, α 1j−1δ m −1α1 j−1δ n 1j−1δ m 1−α1j−1δ n
2.2
for some α 0 ≤ α < 1, m ∈ N, n ∈ N0, δ δ ≥ 0 Then fz ∈ S m,n,δ α.
Proof Suppose that2.1 is true for α0 ≤ α < 1, m ∈ N, n ∈ N0, and δδ ≥ 0 For fz ∈ A, define the function F z by
F z D m f z
It suffices to show that
F F z − 1 z 1 < 1 z ∈ U. 2.4
We note that
F F z − 1 z 1 D D m m f f z/D z/D n n f f z − α − 1 z − α 1
D D m m f f z − 1 αD z 1 − αD n n f f z z
α−∞j21 j − 1δ m − 1 α1 j − 1δ n
a j z j−1
2 − α ∞j21 j − 1δ m 1 − α1 j − 1δ n
a j z j−1
∞
j21 j − 1δ m − 1 α1 j − 1δ n |a j ||z| j−1
2 − α −∞j21 j − 1δ m 1 − α1 j − 1δ n
|a j ||z| j−1
< α∞
j21 j − 1δ m − 1 α1 j − 1δ n |a j|
2 − α −∞j21 j − 1δ m 1 − α1 j − 1δ n
|a j|.
2.5
Trang 3The last expression is bounded above by 1 if
α∞
j21 j − 1δ m − 1 α1 j − 1δ n |a j|
≤ 2 − α −∞
j2
1 j − 1δ m 1 − α1 j − 1δ n
which is equivalent to condition2.1 This completes the proof ofTheorem 2.1
Example 2.2 The function f z given by
f z z ∞
j2
22 γ1 − αj
belongs to the classSm,n,δ α for γ > −2, 0 ≤ α < 1, j ∈ C, and | j| 1
We now derive the coefficient inequalities for fz belonging to the class Sm,n,δ α.
Theorem 2.3 If fz ∈ S m,n,δ α, then for k ≥ 2,
a k ≤ β |v
k|
1 βk−1
j2
1 j − 1δ n
v j β2k−1
j2>j1
k−2
j1 2
1j1− 1δ
1 j2− 1δ n
v j1v j2
β3k−1
j3>j2
k−2
j2>j1
k−3
j1 2
1j1− 1δ 1j2− 1δ 1j3− 1δ n
v j
1v j2v j3 · · ·
β k−2 k−1
j2
1 j − 1δ n
v j| ,
2.8
where β 21 − α and v k 1 k − 1δ m − 1 k − 1δ n
Proof Define the function p z by
p z 1
1− α
D m f z
D n f z − α
1 ∞
j1
Since pz is the Carath´eodory function, we have that
The definition of pz implies that
1
1 − α
D m f z − αD n f z D n f z
1∞
j1
c j z j
Trang 4
D n f z z ∞
j2
1 j − 1δ n
a j z j
n∈ N0
we have
D m f z − αD n f z
1 δ m − α1 δ n
1− α a2z2 1 2δ m − α1 2δ n
1− α a3z3 · · ·
1 k − 1δ m − α1 k − 1δ n
D n f z
1∞
j1
c j z j
z∞
j2
1 j − 1δ n
a j z j
1 c1z · · · c k z k · · ·.
2.13 Therefore,2.11 shows that
z1δ m −α1δ n
1−α a2z212δ m −α12δ n
1−α a3z3· · ·1k−1δ m −α1k−1δ n
z∞
j2
1 j − 1δ n
a j z j
1 c1z · · · c k z k · · ·.
2.14
If we consider the coefficients of zkof the both sides in the above equality, then we find that
1 k − 1δ m − α 1 k − 1δ n
n
a k k−1
j1
1 k − j − 1δ n
a k −j c j
2.15 Therefore,
1 k − 1δ m − 1 k − 1δ n
k−1
j1
1 k − j − 1δ n a k −j c j
≤ 1 k − 1δ 1m − α − 1 k − 1δ n
k−1
j1
1 k − j − 1δ na k −jc j
≤ 1 k − 1δ 21 − αm − 1 k − 1δ n
k−1
j1
1 k − j − 1δ na k −j,
2.16
Trang 5since|c j | ≤ 2 j 1, 2, 3 Thus, for β 21 − α and v k 1 k − 1δ m − 1 k − 1δ n, we obtain
|a k | ≤ β |v1
k|
1 1 δ n β
v2 1 2δ n β
v3 1 3δ n β
v4 ··· 1 k − 2δ n β
v k−1
1 δ n 1 2δ n β2
v2v3 1 δ n 1 3δ n β2
v2v4
1 δ n 1 4δ n β2
v2v5 ··· 1 δ n 1 k − 2δ n β2
v2v k−1
1 2δ n 1 3δ n β2
v3v4 1 2δ n 1 4δ n β2
v3v5 ···
1 2δ n 1 k − 2δ n β2
v3v k−1 ···
1 δ n 1 2δ n 1 3δ n β3
v2v3v4 1 δ n 1 3δ n 1 4δ n β3
v2v4v5 ···
1 δ n 1 k − 3δ n 1 k − 2δ n β3
v2v k−2v k−1 β k−2 k−1
j2
1 j − 1δ n
v j
|v β
k|
1 βk−1
j2
1 j − 1δ n
v j β2k−1
j2>j1
k−2
j1 2
1 j1− 1δ 1 j2− 1δ n
v j1v j2
β3k−1
j3>j2
k−2
j2>j1
k−3
j1 2
1j1−1δ, 1j2−1δ, 1j3−1δ
n
v j
1v j2v j3 · · ·β k−2 k−1
j2
1j−1δ n
v j .
2.17 This completes the proof ofTheorem 2.3
If we take δ 1 in Theorems2.1and2.3, we can get the results due to S ¨umer Eker and Owa3
3 Extreme points
In view ofTheorem 2.1, we now introduce the subclass Sm,n,δ α ⊂ S m,n,δ α, which consists of
function
f z z ∞
j2
whose Taylor-Maclaurin coefficients satisfy inequality 2.1 Now, let us determine extreme points of the class Sm,n,δ α.
Trang 6Theorem 3.1 Let f1z z and
f j z z Ψm, n, j, δ, α21 − α z j j 2, 3, , 3.2
where Ψm, n, j, δ, α is given by 2.2.
Then f ∈ Sm,n α if and only if it can be expressed in the form
f z ∞
j1
where η j > 0 and∞
j1η j 1.
Proof Suppose that
f z ∞
j1
η j f j z z ∞
j2
η j 21 − α
Then
∞
j2
Ψm, n, j, δ, α Ψm, n, j, δ, α21 − α η j 21 − α∞
j2
η j 21 − α1 − η1 < 21 − α, 3.5
which shows that f satisfies condition2.1 and therefore f ∈ Sm,n,δ α.
Conversely, suppose that f ∈ Sm,n,δ α Since
a j ≤ Ψm, n, j, δ, α21 − α j 2, 3, , 3.6
we may set
η j Ψm, n, j, δ, α21 − α a j ,
η1 1 −∞
j2
η j
3.7
Then we obtain
f z ∞
j1
which completes the proof ofTheorem 3.1
Corollary 3.2 The extreme points of Sm,n,δ α are the functions f1z z and
f j z z Ψm, n, j, δ, α21 − α z j j 2, 3, , 3.9
where Ψm, n, j, δ, α is given by 2.2.
Trang 74 Integral means inequalities for fractional derivative
We will make use of the following definitions of fractional derivatives by Owa4 , and Srivas-tava and Owa5
Definition 4.1 The fractional derivative of order λ is defined, for a function f, by
D λ z f z Γ1 − λ1 d
dz
z
0
f ξ
where f is an analytic function in a simply connected region of z-plane containing the origin,
and the multiplicity ofz − ξ −λis removed by requiring logz − ξ to be real when z − ξ > 0
Definition 4.2 Under the hypotheses ofDefinition 4.1, the fractional derivative of order p λ is defined, for a function f, by
D p z λ f z d p
dz p D z λ f z 0 ≤ λ < 1; p ∈ N0. 4.2
It readily follows from4.1 that
D z λ z k Γk − λ 1 Γk 1 z k −λ 0 ≤ λ < 1, k ∈ N. 4.3
Further, we need the concept of subordination between analytic functions 6 and a subordi-nation theorem of Littlewood in our investigation
Definition 4.3 For two functions f and g, analytic in U, say that the function f z is subordinate
to g z in U, and write
if there exists a Schwarz function wz, analytic in U with w0 0 and |wz| < 1 such that
In particular, if the function g is univalent inU, the above subordination is equivalent to
In 1925, Littlewood7 proved the following subordination theorem
Lemma 4.4 If fz and gz are analytic in U with fz≺g(z), then for μ>0 and zre iθ 0<r <1,
2π
0
|fz| μ dθ
2π
0
Trang 8Theorem 4.5 Let fz ∈ Sm,n,δ α and suppose that
∞
j2
j − p p1a j ≤ Ψm, n, k, δ, αΓk 1 − λ − pΓ2 − p21 − αΓk 1Γ3 − λ − p 4.8
for some j ≥ p, 0 ≤ λ < 1, and j − p p1 denote the Pochhammer symbol defined by j − p p1
j − pj − p 1 · · · j Also let the function
f k z z Ψm, n, k, δ, α21 − α z k k ≥ 2. 4.9
If there exists an analytic function w z given by
wz k−1 Ψm, n, k, δ, αΓk 1 − λ − p21 − αΓk 1 ∞
j2
j − p p1Γj 1 − λ − p Γj − p a j z j−1, k ≥ p,
4.10
then for z re iθ and 0 < r < 1,
2π
0
D p λ
z f zμ
dθ≤
2π
0
D p λ
z f k zμ
dθ 0 ≤ λ < 1, μ > 0. 4.11
Proof By virtue of the fractional derivative formula4.3 andDefinition 4.2, we find from3.1 that
D p z λ f z Γ2 − λ − p z1−λ−p
1∞
j2
Γ2 − λ − pΓj 1
Γj 1 − λ − p a j z j−1
Γ2 − λ − p
1∞
j2
Γ2 − λ − pj − p p1Φja j z j−1 ,
4.12
where
Φj Γj 1 − λ − p Γj − p 0 ≤ λ < 1; j ≥ p. 4.13 SinceΦj is a decreasing function of j, we have
0 < Φj ≤ Φ2 Γ3 − λ − p Γ2 − p 4.14 Similarly, from4.3, 4.9, andDefinition 4.2, we obtain
D p z λ f k z z1−λ−p
Γ2 − λ − p
1 21 − αΓ2 − λ − pΓk 1
Ψm, n, k, δ, αΓk 1 − λ − p z k−1
Trang 9
For z re iθ , 0 < r < 1, we must show that
2π
0
1
∞
j2
Γ2 − λ − pj − p p1Φja j z j−1
μ
dθ
≤
2π
0
1 Ψm, n, k, δ, αΓk 1 − λ − p21 − αΓ2 − λ − pΓk 1 z k−1
μ dθ μ > 0.
4.16
Thus by applying Littlewood’s subordination theorem, it would be suffice to show that
1∞
j2
Γ2 − λ − pj − p p1Φja j z j−1≺ 1 Ψm, n, k, δ, αΓk 1 − λ − p21 − αΓ2 − λ − pΓk 1 z k−1. 4.17
By setting
1∞
j2
Γ2 − λ − pj − p p1Φja j z j−1 1 21 − αΓ2 − λ − pΓk 1
Ψm, n, k, δ, αΓk 1 − λ − p w z k−1, 4.18
we find that
wz k−1 Ψm, n, k, δ, αΓk 1 − λ − p21 − αΓk 1 ∞
j2
j − p p1Φja j z j−1 4.19
which readily yields w0 0 Further, we prove that the analytic function wz satisfies
|wz| < 1, z ∈ U for 4.10 We know that
|wz| k−1≤
Ψm, n, k, δ, αΓk 1 − λ − p
21 − αΓk 1
∞
j2
j − p p1Φja j z j−1
≤ Ψm, n, k, δ, αΓk 1 − λ − p
21 − αΓk 1
∞
j2
j − p p1Φja j |z| j−1
≤ |z| Ψm, n, k, δ, αΓk 1 − λ − p
∞
j2
j − p p1a j
|z| Ψm, n, k, δ, αΓk 1 − λ − p21 − αΓk 1 Γ3 − λ − p Γ2 − p ∞
j2
j − p p1a j
≤ |z| < 1
4.20
by means of the hypothesis ofTheorem 4.5
As special case p 0,Theorem 4.5readily yields
Corollary 4.6 Let fz ∈ Sm,n,δ α and suppose that
∞
j2
ja j ≤ Ψm, n, k, δ, αΓk 1 − λ21 − αΓk 1Γ3 − λ 4.21
Trang 10for some 0 ≤ λ < 1 Also let the function
f k z z Ψm, n, k, δ, α21 − α z k k ≥ 2. 4.22
If there exists an analytic function w z given by
wz k−1 Ψm, n, k, δ, αΓk 1 − λ21 − αΓk 1 ∞
j2
Γj 1
Γj 1 − λ a j z j−1, 4.23
then for z re iθ and 0 < r < 1,
2π
0
D λ
z f zμ
dθ≤
2π
0
D λ
z f k zμ
dθ 0 ≤ λ < 1, μ > 0. 4.24
Acknowledgment
The authors are thankful to the referees for their comments and suggestions
References
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... Trang 74 Integral means inequalities for fractional derivative
We will make use of the following...
j1
c j z j
Trang 4
D...
0
Trang 8Theorem 4.5 Let fz ∈ Sm,n,δ α and suppose that
∞