Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 390435, 14 pages
doi:10.1155/2008/390435
Research Article
New Classes of Analytic Functions Involving
Generalized Noor Integral Operator
Rabha W. Ibrahim and Maslina Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Selangor Darul Ehsan, Bangi 43600, Malaysia
Correspondence should be addressed to Maslina Darus, maslina@pkrisc.cc.ukm.my
Received 22 March 2008; Accepted 25 April 2008
Recommended by Jozsef Szabados
The present article investigates new classes of functions involving generalized Noor integral
operator. Some properties of these functions are studied including characterization and distortion
theorems. Moreover, we illustrate sufficient conditions for subordination and superordination for
analytic functions.
Copyright q 2008 R. W. Ibrahim and M. Darus. 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 preliminaries
Let H be the class of functions analytic in U and let Ha, n be the subclass of H consisting of
functions of the form fz a an zn an1 zn1 · · · . Let A be the subclass of H consisting of
functions of the form fz z a2 z2 · · · .
Denote by Dα : A → A the operator defined by
Dα :
z
1 − zα1
∗fz,
α > −1,
1.1
where ∗ refers to the Hadamard product or convolution. Then implies that
n
z zn−1 fz
D fz ,
n!
n
n ∈ N0 N ∪ {0}.
1.2
We note that D0 fz fz and D fz zf z. The operator Dn f is called Ruscheweyh
derivative of nth order of f. Noor 1, 2 defined and studied an integral operator In : A → A
analogous to Dn f as follows.
2
Journal of Inequalities and Applications
−1
Let fn z z/1 − zn1 , n ∈ N0 , and let fn
−1
fn z∗fn
z be defined such that
z
.
1−z
1.3
Then
In fz −1
fn z∗fz
z
1 − zn1
−1
∗fz.
1.4
Note that I0 fz zf z and I1 fz fz. The operator In is called the Noor Integral of nth
order of f. Using 1.3, 1.4, and a well-known identity for Dn f, we have
1.5
n 1In fz − nIn1 fz z In1 fz .
Using hypergeometric functions 2 F1 , 1.4 becomes
In fz z 2 F1 1, 1; n 1, z ∗fz,
1.6
where 2 F1 a, b; c, z is defined by
2 F1 a, b; c, z
1
ab z aa 1bb 1 z2
··· .
c 1!
cc 1
2!
1.7
For complex parameters
αj
0, −1, −2, . . . ; j 1, . . . , q ,
/
Aj
βj
β1 , . . . , βp
0,
−1,
−2,
.
.
.
;
j
1,
.
.
.
,
p
,
/
Bj
α1 , . . . , αq
1.8
the Fox-Wright generalization q Ψp z of the hypergeometric q Fp function bysee 3–5
⎡
q Ψp
⎢
⎢
⎣
α1 , A1 , . . . , αq , Aq ;
β1 , B1 , . . . , βp , Bp ;
⎤
⎥
z⎥
⎦ q Ψp αj , Aj 1,q ; βj , Bj 1,p ; z
∞ Γ α nA · · · Γ α nA
n
1
1
q
q z
:
n!
n0 Γ β1 nB1 · · · Γ βp nBp
q
∞
n
j1 Γ αj nAj z
,
p n!
n0
j1 βj nBj
1.9
p
q
where Aj > 0 for all j 1, . . . , q, Bj > 0 for all j 1, . . . , p, and 1 j1 Bj − j1 Aj ≥ 0 for
suitable values |z|. For special case, when Aj 1 for all j 1, . . . , q, and Bj 1 for all j 1, . . . , p,
we have the following relationship:
q Fp α1 , . . . , αq ; β1 , . . . , βp ; z Ω q Ψp αj , 1 1,q ; βj , 1 1,p ; z ,
1.10
q ≤ p 1; q, p ∈ N0 N ∪ {0}, z ∈ U,
R. W. Ibrahim and M. Darus
3
where
Γ β1 · · · Γ βp
Ω : .
Γ α1 · · · Γ αq
1.11
We introduce a function z q Ψp αj , Aj 1,q ; βj , Bj 1,p ; z−1 given by
−1
z q Ψp αj , Aj 1,q ; βj , Bj 1,p ; z ∗ zq Ψp αj , Aj 1,q ; βj , Bj 1,p ; z
z
1 − z
λ1
z
∞
λ 1n−1
n2
n − 1!
zn ,
1.12
λ > −1,
and obtain the following linear operator:
−1
Iλ αj , Aj 1,q ; βj , Bj 1,p fz z q Ψp αj , Aj 1,q ; βj , Bj 1,p ; z
∗fz,
1.13
where f ∈ A, z ∈ U, and
z q Ψp αj , Aj
1,q
; βj , Bj
1,p
;z
−1
z
p
j1 Γ βj
q n2
j1 Γ αj
∞
n − 1Bj
n − 1Aj
λ 1n−1 zn .
1.14
For some computation, we have
Iλ
p ∞
j1 Γ βj n − 1Bj
αj , Aj 1,q ; βj , Bj 1,p fz z λ 1n−1 an zn ,
q Γ
α
n
−
1A
j
j
n2
j1
1.15
where an is the Pochhammer symbol defined by
⎧
n0
Γa n ⎨1,
an ⎩
Γa
aa 1 · · · a n − 1, n {1, 2, . . .}.
1.16
From 1.15 we have the following result.
Lemma 1.1. Let fz ∈ A for all z ∈ U then
i I0 1, 11,1 ; 1, 1/n − 11,p fz fz.
ii I1 1, 11,1 ; 1, 1/n − 11,p fz zf z.
iii zIλ αj , Aj 1,q ; βj , Bj 1,p fz λ1Iλ1 αj , Aj 1,q ; βj , Bj 1,p fz−λIλ αj , Aj 1,q ;
βj , Bj 1,p fz.
In the following definitions, we introduce new classes of analytic functions containing
generalized Noor integral operator 1.15.
4
Journal of Inequalities and Applications
μ
Definition 1.2. Let fz ∈ A then fz ∈ Sλ αj , Aj 1,q ; βj , Bj 1,p if and only if
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
R
> μ,
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
0 ≤ μ < 1, z ∈ U.
1.17
μ
Definition 1.3. Let fz ∈ A then fz ∈ Cλ αj , Aj 1,q ; βj , Bj 1,p if and only if
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz > μ,
R
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
0 ≤ μ < 1, z ∈ U.
1.18
Let F and G be analytic functions in the unit disk U. The function F is subordinate to G,
written F ≺ G, if G is univalent, F0 G0 and FU ⊂ GU. Or given two functions Fz
and Gz, which are analytic in U, the function Fz is said to be subordination to Gz in U if
there exists a function hz, analytic in U with
h0 0,
|hz| < 1
∀z ∈ U,
1.19
such that
Fz G hz
∀z ∈ U.
1.20
Let φ : C2 → C and let h be univalent in U. If p is analytic in U and satisfies the
differential subordination φpz, zp z ≺ hz then p is called a solution of the differential
subordination. The univalent function q is called a dominant of the solutions of the differential
subordination, p ≺ q. If p and φpz, zp z are univalent in U and satisfy the differential
superordination hz ≺ φpz, zp z then p is called a solution of the differential
superordination. An analytic function q is called subordinant of the solution of the differential
superordination if q ≺ p. Let Φ be an analytic function in a domain containing fU, Φ0 0
and Φ 0 > 0.
The function f ∈ A is called Φ—like if
R
zf z
Φ fz
> 0,
z ∈ U.
1.21
This concept was introduced by Brickman 6 and established that a function f ∈ A is
univalent if and only if f is Φ—like for some Φ.
Definition 1.4. Let Φ be analytic function in a domain containing fU, Φ0 0, Φ 0 1,
and Φω /
0 for ω ∈ fU − 0. Let qz be a fixed analytic function in U, q0 1. The function
f ∈ A is called Φ—like with respect to q if
zf z
≺ qz,
Φ fz
z ∈ U.
1.22
R. W. Ibrahim and M. Darus
5
In the present paper, we apply a method based on the differential subordination in order
to obtain subordination results involving generalized Noor integral operator for a normalized
analytic function fz z ∈ U
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
q1 z ≺ ≺ q2 z.
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
1.23
In order to prove our subordination and superordination results, we need to the following
lemmas in the sequel.
Definition 1.5 see 7. Denote by Q the set of all functions fz that are analytic and injective
on U − Ef, where Ef : {ζ ∈ ∂U : limz→ζ fz ∞} and are such that f ζ / 0 for ζ ∈
∂U − Ef.
Lemma 1.6 see 8. Let qz be univalent in the unit disk U and θ and let φ be analytic in a
domain D containing qU with φw /
0, when w ∈ qU. Set Qz : zq zφqz, hz :
θqz Qz. Suppose that
1 Qz is starlike univalent in U,
2 Rzh z/Qz > 0 for z ∈ U.
If
θ pz zp zφ pz ≺ θ qz zq zφ qz ,
1.24
pz ≺ qz,
1.25
then
and qz is the best dominant.
Lemma 1.7 9. Let qz be convex univalent in the unit disk U and let ϑ and ϕ be analytic in a
domain D containing qU. Suppose that
1 zq zϕqz is starlike univalent in U,
2 R{ϑ qz/ϕqz} > 0 for z ∈ U.
If pz ∈ Hq0, 1 ∩ Q, with pU ⊆ D and ϑpz zp zϕz being univalent in U and
ϑ qz zq zϕ qz ≺ ϑ pz zp zϕ pz ,
1.26
qz ≺ pz,
1.27
then
and qz is the best subordinant.
6
Journal of Inequalities and Applications
2. Characterization properties and distortion theorems
In this section, we investigate the characterization properties for the function fz ∈ A
μ
μ
to belong to the classes Sλ αj , Aj 1,q ; βj , Bj 1,p and Cλ αj , Aj 1,q ; βj , Bj 1,p by obtaining
μ
the coefficient bounds. Further, we prove the distortion theorems when fz ∈ Sλ αj ,
μ
Aj 1,q ; βj , Bj 1,p and fz ∈ Cλ αj , Aj 1,q ; βj , Bj 1,p .
μ
Theorem 2.1. Let fz ∈ A. Then fz ∈ Sλ αj , Aj 1,q ; βj , Bj 1,p if and only if
∞
Hn−1 an μλ 1n−1 − λ 1n − λn ≤ 1 − μ,
0 ≤ μ < 1,
2.1
n2
where
p
Hn−1 :
j1 Γ βj
q j1 Γ αj
n − 1Bj
n − 1Aj
2.2
.
Proof. Suppose that 2.1 holds. Then by using Lemma 1.1 and for z ∈ U, we have
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
R
≤
Iλ αj , Aj
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
;
β
,
B
fz
j
j 1,p
1,q
≤
1
n2 Hn−1 an λ 1n − λn
.
1 ∞
n2 Hn−1 an λ 1n−1
∞
2.3
μ
This last expression is greater than μ, if 2.1 holds this implies that fz ∈ Sλ αj ,
μ
Aj 1,q ; βj , Bj 1,p . On the other hand, assume that fz ∈ Sλ αj , Aj 1,q ; βj , Bj 1,p then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
R
> μ,
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
2.4
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz > μ.
Iλ αj , Aj
; βj , Bj
fz 2.5
but R{z} ≤ |z| then
1,q
1,p
By a computation, we obtain 2.1.
μ
Corollary 2.2. Let the function fz belong to the class Sλ αj , Aj 1,q ; βj , Bj 1,p . Then
an ≤
1 − μ
,
Hn−1 μλ 1n−1 − λ 1n − λn where Hn−1 is defined in 2.2.
0 ≤ μ < 1,
2.6
R. W. Ibrahim and M. Darus
7
μ
Theorem 2.3. Let fz ∈ A. Then fz ∈ Cλ αj , Aj 1,q ; βj , Bj 1,p if and only if
∞
nHn−1 an μλ 1n−1 − λ 1n − λn ≤ 1 − μ,
0 ≤ μ < 1,
2.7
n2
where Hn−1 is defined in 2.2.
μ
Corollary 2.4. Let the function fz belong to the class Cλ αj , Aj 1,q ; βj , Bj 1,p . Then
an ≤
nHn−1 μλ 1
1 − μ
,
n−1 − λ 1n − λn
0 ≤ μ < 1,
2.8
where Hn−1 is defined in 2.2.
μ
Theorem 2.5. Let fz ∈ Sλ αj , Aj 1,q ; βj , Bj 1,p , then
fz ≥ |z| −
1 − μ
|z|2 ,
H1 μλ 11 − λ 12 − λ2 fz ≤ |z| 1 − μ
|z|2 ,
H1 μλ 11 − λ 12 − λ2 2.9
for z ∈ U where Hn−1 is defined in 2.2.
μ
Proof. If fz ∈ Sλ αj , Aj 1,q ; βj , Bj 1,p then in view of Theorem 2.1, we have
∞ ∞
H1 μλ 11 − λ 12 − λ2 an ≤
Hn−1 an μλ 1n−1 − λ 1n − λn n2
n2
2.10
≤ 1 − μ.
This yields
∞ an ≤
n2
1−μ
.
H1 μλ 11 − λ 12 − λ2 2.11
Now
∞ fz ≥ |z| − |z|2 an n2
1 − μ
≥ |z| −
|z|2 .
H1 μλ 11 − λ 12 − λ2 2.12
Also,
fz ≤ |z| Hence the proof is complete.
1 − μ
|z|2 .
H1 μλ 11 − λ 12 − λ2 2.13
8
Journal of Inequalities and Applications
Corollary 2.6. Under the hypothesis of Theorem 2.5, fz is included in a disk with its center at the
origin and radius r given by
1 − μ
.
H1 μλ 11 − λ 12 − λ2 r 1
2.14
In the same way, we can prove the following result.
μ
Theorem 2.7. Let fz ∈ Cλ αj , Aj 1,q ; βj , Bj 1,p then
fz ≥ |z| −
1 − μ
|z|2 ,
2H1 μλ 11 − λ 12 − λ2 fz ≤ |z| 1 − μ
|z|2 ,
2H1 μλ 11 − λ 12 − λ2 2.15
for z ∈ U where Hn−1 is defined in 2.2.
Corollary 2.8. Under the hypothesis of Theorem 2.7, fz is included in a disk with its center at the
origin and radius r given by
r 1
1 − μ
.
2H1 μλ 11 − λ 12 − λ2 μ
2.16
μ
We next study some properties of the classes Sλ αj , Aj 1,q ; βj , Bj 1,p and Cλ αj , Aj 1,q ;
βj , Bj 1,p .
Theorem 2.9. Let λ > −1 and 0 ≤ μ1 < μ2 < 1. Then
μ
Sλ 2
αj , Aj
1,q
μ ; βj , Bj 1,p ⊂ Sλ1 αj , Aj 1,q ; βj , Bj 1,p .
2.17
Proof. By using Theorem 2.1.
Theorem 2.10. Let −1 < λ1 ≤ λ2 and 0 ≤ μ < 1. Then
μ
Sλ1
αj , Aj
1,q
μ ; βj , Bj 1,p ⊇ Sλ2 αj , Aj 1,q ; βj , Bj 1,p .
2.18
Proof. By using Theorem 2.1.
Theorem 2.11. Let λ > −1 and 0 ≤ μ1 < μ2 < 1. Then
μ
Cλ 2
αj , Aj
1,q
μ ; βj , Bj 1,p ⊂ Cλ 1 αj , Aj 1,q ; βj , Bj 1,p .
2.19
Proof. By using Theorem 2.3.
Theorem 2.12. Let −1 < λ1 ≤ λ2 and 0 ≤ μ < 1. Then
μ
Cλ1
αj , Aj
Proof. By using Theorem 2.3.
1,q
μ ; βj , Bj 1,p ⊇ Cλ2 αj , Aj 1,q ; βj , Bj 1,p .
2.20
R. W. Ibrahim and M. Darus
9
3. Sandwich results
By making use of Lemmas 1.6 and 1.7, we prove the following subordination and
superordination results.
Theorem 3.1. Let qz /
0 be univalent in U such that zq z/qz is starlike univalent in U and
zq z zq z
α
> 0,
R 1 qz −
γ
q z
qz
α, γ ∈ C, γ /
0.
3.1
If f ∈ A satisfies the subordination
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
α
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.2
γ 1 −
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ αqz γzq z
,
qz
then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ qz,
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.3
and qz is the best dominant.
Proof. Our aim is to apply Lemma 1.6. Setting
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
pz : .
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.4
Computation shows that
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zp z
1 ,
−
pz
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.5
which yields the following subordination:
αpz γzp z
γzq z
≺ αqz ,
pz
qz
α, γ ∈ C.
3.6
10
Journal of Inequalities and Applications
By setting
θω : αω,
φω :
γ
,
ω
γ
/ 0,
3.7
it can be easily observed that θω is analytic in C and φω is analytic in C \ {0} and that
φω /
0 when ω ∈ C \{0}. Also, by letting
q z
Qz zq zφ qz γz
,
qz
q z
hz θ qz Qz αqz γz
,
qz
we find that Qz is starlike univalent in U and that
zq z zq z
α
zh z
1 qz −
> 0.
R
Qz
γ
q z
qz
3.8
3.9
Then the relation 3.3 follows by an application of Lemma 1.6.
Corollary 3.2. Let the assumptions of Theorem 2.1 hold. Then the subordination
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
α − γ γ 1 Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ αqz γzq z
,
qz
3.10
implies
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ qz,
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.11
and qz is the best dominant.
Proof. By letting Φω : ω.
Corollary 3.3. If f ∈ A and assume that 3.1 holds then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
A − Bz
1 Az
1 ≺
1
Bz
1
Az1 Bz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.12
implies
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
1 Az
,
≺
1 Bz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
and 1 Az/1 Bz is the best dominant.
−1 ≤ B < A ≤ 1,
3.13
R. W. Ibrahim and M. Darus
11
Proof. By setting Φω : ω, α γ 1, and qz : 1Az/1Bz, where −1 ≤ B < A ≤ 1.
Corollary 3.4. If f ∈ A and assume that 3.1 holds then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
2z
1z
1 ≺
1
−
z
1
− z2
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.14
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
1z
,
≺
1−z
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.15
implies
and 1 z/1 − z is the best dominant.
Proof. By setting Φω : ω, α γ 1, and qz : 1 z/1 − z.
Corollary 3.5. If f ∈ A and assume that 3.1 holds then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Az
1 ≺ e Az
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.16
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ eAz ,
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.17
implies
and eAz is the best dominant.
Proof. By setting Φω : ω, α γ 1, and qz : eAz , |A| < π.
Theorem 3.6. Let qz /
0 be convex univalent in the unit disk U. Suppose that
α
qz
R
γ
> 0,
α, γ ∈ C for z ∈ U,
3.18
and that zq z/qz is starlike univalent in U. If zIλ αj , Aj 1,q ; βj , Bj 1,p fz /ΦIλ αj ,
Aj 1,q ; βj , Bj 1,p fz ∈ Hq0, 1 ∩ Q where f ∈ A,
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
α
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γ 1 −
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.19
12
Journal of Inequalities and Applications
is univalent in U and the subordination
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γzq z
αqz ≺α
qz
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γ 1 Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
−
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.20
holds, then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
qz ≺ Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.21
and q is the best subordinant.
Proof. Our aim is to apply Lemma 1.7. Setting
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
pz : .
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.22
Computation shows that
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zp z
1 ,
−
pz
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.23
which yields the following subordination
αqz γzq z
γzp z
≺ αpz ,
qz
pz
α, γ ∈ C.
3.24
By setting
ϑω : αω,
ϕω :
γ
,
ω
γ/
0,
3.25
it can be easily observed that θω is analytic in C and φω is analytic in C \ {0}, and that
φω / 0 when ω ∈ C \ {0}. Also, we obtain
ϑ qz
α
R
qz > 0.
R
γ
ϕ qz
Then 3.21 follows by an application of Lemma 1.7.
Combining Theorems 3.1 and 3.6 in order to get the following sandwich theorems
3.26
R. W. Ibrahim and M. Darus
13
Theorem 3.7. Let q1 z /
0, q2 z /
0 be convex univalent in the unit disk U satisfying 3.18 and
3.1, respectively. Suppose that zqi z/qi z, i 1, 2, is starlike univalent in U. If
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
∈ H q0, 1 ∩ Q,
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.27
where f ∈ A,
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
α
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γ 1 −
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.28
is univalent in U, and the subordination
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γzq1 z
≺α
αq1 z q1 z
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
γ 1 Iλ αj , Aj 1,q ; βj , Bj 1,p fz
zΦ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
−
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
≺ αq2 z 3.29
γzq2 z
q2 z
holds, then
z Iλ αj , Aj 1,q ; βj , Bj 1,p fz
q1 z ≺ ≺ q2 z,
Φ Iλ αj , Aj 1,q ; βj , Bj 1,p fz
3.30
and q1 z is the best subordinant and q2 z is the best dominant.
Acknowledgment
The work presented here was supported by SAGA: STGL-012-2006, Academy of Sciences,
Malaysia.
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