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Ibrahim and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor Darul Ehsan, Bangi 43600, Malaysia Correspondence s

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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

LetH 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 a n z n a n1 z n1 · · · Let A be the subclass of H consisting of functions of the form fz z a2z2 · · ·

Denote by D α:A → A the operator defined by

D α: z

where∗ refers to the Hadamard product or convolution Then implies that

D n fz z

z n−1 fzn

n! , n ∈ N0 N ∪ {0}. 1.2

We note that D0fz fz and Dfz zfz The operator D n f is called Ruscheweyh

derivative of nth order of f Noor 1,2 defined and studied an integral operator I n : A → A

analogous to D n f as follows.

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Let f n z z/1 − z n1 , n ∈ N0, and let f n−1be defined such that

f n z∗f n−1z z

Then

I n fz f n−1z∗fz 

z

1 − z n1

−1

Note that I0fz zfz and I1fz fz The operator I n is called the Noor Integral of nth order of f Using 1.3, 1.4, and a well-known identity for D n f, we have

n 1I n fz − nI n1 fz z

I n1 fz

Using hypergeometric functions2F1, 1.4 becomes

I n fz 

where2F1a, b; c, z is defined by

2F1a, b; c, z 1 ab

c

z

1! aa 1bb 1

cc 1

z2

For complex parameters

α1, , α q

α j

A j / 0, −1, −2, ; j 1, , q

,

β1, , β p

β

j

B j / 0, −1, −2, ; j 1, , p ,

1.8

the Fox-Wright generalizationqΨp z of the hypergeometric q F p function bysee 3 5

qΨp

⎡

⎢

⎣

α1, A1

, ,

α q , A q

;

z

β1, B1

, ,

β p , B p

;

⎤

⎥

⎦ qΨpα j , A j

1,q;

β j , B j

1,p ; z

:∞

n0

Γα1 nA1

· · · Γα q nA q

Γβ1 nB1

· · · Γβ p nB p

z n

n!

∞

n0

q j1Γα j nA j

p j1

β j nB j

z n

n! ,

1.9

where A j > 0 for all j 1, , q, B j > 0 for all j 1, , p, and 1 p

j1 B j −q

j1 A j ≥ 0 for suitable values|z| For special case, when A j 1 for all j 1, , q, and B j 1 for all j 1, , p,

we have the following relationship:

q F p

α1, , α q ; β1, , β p ; z

ΩqΨp

α j , 1

1,q;

β j , 1

1,p ; z

,

q ≤ p 1; q, p ∈ N0 N ∪ {0}, z ∈ U, 1.10

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Ω : Γ

β1

· · · Γβ p

Γα1

· · · Γα q

We introduce a functionz qΨp α j , A j1,q;β j , B j1,p ; z−1given by

z qΨp

α j , A j

1,q;

β j , B j

1,p ; z

∗z qΨp

α j , A j

1,q;

β j , B j

1,p ; z−1

1 − z λ1 z ∞

n2

λ 1 n−1

n − 1! z n , λ > −1,

1.12

and obtain the following linear operator:

I λ

α j , A j

1,q;

β j , B j

1,p

fz 

z qΨpα j , A j

1,q;

β j , B j

1,p ; z−1

where f ∈ A, z ∈ U, and

z qΨp

α j , A j

1,q;

β j , B j

1,p ; z−1

 z ∞

n2

p j1Γβ j n − 1B j

q j1Γα j n − 1A j λ 1 n−1 z n 1.14 For some computation, we have

I λ

α j , A j

1,q;

β j , B j

1,p

fz z 

∞

n2

p j1Γβ j n − 1B j

q j1Γα j n − 1A j λ 1 n−1 a n z n , 1.15 wherea nis the Pochhammer symbol defined by

a n Γa n Γa

⎧

⎨

⎩

aa 1 · · · a n − 1, n {1, 2, }. 1.16 From1.15 we have the following result

Lemma 1.1 Let fz ∈ A for all z ∈ U then

i I01, 1 1,1;1, 1/n − 1 1,p fz fz.

ii I11, 1 1,1;1, 1/n − 1 1,p fz zfz.

iii zI λ α j , A j1,q;β j , B j1,p fz λ1I λ1 α j , A j1,q;β j , B j1,p fz−λI λ α j , A j1,q;

β j , B j1,p fz.

In the following definitions, we introduce new classes of analytic functions containing generalized Noor integral operator1.15

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Definition 1.2 Let fz ∈ A then fz ∈ S μ λ α j , A j1,q;β j , B j1,p if and only if

R

z

I λ

α j , A j

1,q;

β j , B j

1,p fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

> μ, 0≤ μ < 1, z ∈ U. 1.17

Definition 1.3 Let fz ∈ A then fz ∈ C μ λ α j , A j1,q;β j , B j1,p if and only if

R

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

> μ, 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

Let φ : C2→ C and let h be univalent in U If p is analytic in U and satisfies the

differential subordination φpz, zpz ≺ 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, zpz are univalent in U and satisfy the differential superordination hz ≺ φpz, zpz 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

andΦ0 > 0.

The function f ∈ A is called Φ—like if

R

zfz

Φfz

> 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, Φ0 1,

andΦω / 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

zfz

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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

q1z ≺ z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j1,q;

β j , B j

1,p

In order to prove our subordination and superordination results, we need to the following lemmas in the sequel

Definition 1.5see 7 Denote by Q the set of all functions fz that are analytic and injective

on U − Ef, where Ef : {ζ ∈ ∂U : lim z→ζ fz ∞} and are such that fζ / 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 : zqzφqz, hz :

θqz Qz Suppose that

1 Qz is starlike univalent in U,

2 Rzhz/Qz > 0 for z ∈ U.

If

θ

pz

 zpzφpz

≺ θqz

 zqzφqz

then

pz ≺ qz, 1.25

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 zqzϕqz is starlike univalent in U,

2 R{ϑqz/ϕqz} > 0 for z ∈ U.

If pz ∈ Hq0, 1 ∩ Q, with pU ⊆ D and ϑpz zpzϕz being univalent in U and

ϑ

qz

 zqzϕqz

≺ ϑpz

 zpzϕpz

then

qz ≺ pz, 1.27

and qz is the best subordinant.

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2 Characterization properties and distortion theorems

In this section, we investigate the characterization properties for the function fz ∈ A

to belong to the classes S μ λ α j , A j1,q;β j , B j1,p and C μ λ α j , A j1,q;β j , B j1,p by obtaining the coeﬃcient bounds Further, we prove the distortion theorems when fz ∈ Sμ

λ α j ,

A j1,q;β j , B j1,p and fz ∈ C μ

λ α j , A j1,q;β j , B j1,p .

Theorem 2.1 Let fz ∈ A Then fz ∈ S μ

λ α j , A j1,q;β j , B j1,p if and only if

∞

n2

H n−1a nμλ 1 n−1−

λ 1 n − λ n ≤ 1 − μ, 0 ≤ μ < 1, 2.1

where

H n−1:

p j1Γβ j n − 1B j

q j1Γα j n − 1A j

Proof Suppose that2.1 holds Then by usingLemma 1.1and for z ∈ U, we have

R

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

≤

z

I λ

α j , A j

1,q;

β j , B j

1,p fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

≤ 1

∞

n2 H n−1a n λ 1 n − λ n

1∞n2 H n−1a n λ 1 n−1

2.3

This last expression is greater than μ, if 2.1 holds this implies that fz ∈ S μ λ α j ,

A j1,q;β j , B j1,p On the other hand, assume that fz ∈ S μ λ α j , A j1,q;β j , B j1,p then

R

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

> μ, 2.4

butR{z} ≤ |z| then

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

By a computation, we obtain2.1

Corollary 2.2 Let the function fz belong to the class S μ

λ α j , A j1,q;β j , B j1,p Then

H n−1μλ 1 n−1−

λ 1 n − λ n , 0 ≤ μ < 1, 2.6

where H n−1 is defined in2.2.

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Theorem 2.3 Let fz ∈ A Then fz ∈ C μ

λ α j , A j1,q;β j , B j1,p if and only if

∞

n2

nH n−1a nμλ 1 n−1−

λ 1 n − λ n ≤ 1 − μ, 0 ≤ μ < 1, 2.7

Corollary 2.4 Let the function fz belong to the class C μ

λ α j , A j1,q;β j , B j1,p Then

nH n−1μλ 1 n−1−

λ 1 n − λ n , 0 ≤ μ < 1, 2.8

Theorem 2.5 Let fz ∈ S μ

λ α j , A j1,q;β j , B j1,p , then

fz ≥ |z| − 1 − μ

H1μλ 11−

λ 12− λ2|z|2,

fz ≤ |z| 1 − μ

H1μλ 11−

λ 12− λ2|z|2,

2.9

for z ∈ U where H n−1 is defined in2.2.

Proof If fz ∈ S μ λ α j , A j1,q;β j , B j1,p then in view ofTheorem 2.1, we have

H1μλ 11−

λ 12− λ2∞

n2

a n ≤∞

n2

H n−1a nμλ 1 n−1−

λ 1 n − λ n

≤ 1 − μ.

2.10

This yields

∞

n2

H1μλ 11−

Now

fz ≥ |z| − |z|2∞

n2

a n

H1μλ 11−

λ 12− λ2|z|2.

2.12

Also,

fz ≤ |z| 1 − μ

H1μλ 11−

λ 12− λ2|z|2

Hence the proof is complete

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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

r 1 1 − μ

H1μλ 11−

In the same way, we can prove the following result

Theorem 2.7 Let fz ∈ C μ

λ α j , A j1,q;β j , B j1,p then

2H1μλ 11−

λ 12− λ2|z|2,

2H1μλ 11−

λ 12− λ2|z|2,

2.15

for z ∈ U where H n−1 is defined in2.2.

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

2H1μλ 11−

We next study some properties of the classes S μ λ α j , A j1,q;β j , B j1,p and C μ

λ α j , A j1,q;

β j , B j1,p .

Theorem 2.9 Let λ > −1 and 0 ≤ μ1< μ2< 1 Then

S μ2

λ

α j , A j

1,q;

β j , B j

1,p

⊂ S μ1

λ

α j , A j

1,q;

β j , B j

1,p

Proof By usingTheorem 2.1

Theorem 2.10 Let −1 < λ1≤ λ2and 0 ≤ μ < 1 Then

S μ λ1

α j , A j

1,q;

β j , B j

1,p

⊇ S μ λ2α j , A j

1,q;

β j , B j

1,p

Theorem 2.11 Let λ > −1 and 0 ≤ μ1< μ2< 1 Then

C μ2

λ

α j , A j

1,q;

β j , B j

1,p

⊂ C μ1

λ

α j , A j

1,q;

β j , B j

1,p

Theorem 2.12 Let −1 < λ1≤ λ2and 0 ≤ μ < 1 Then

C λ μ1

α j , A j

1,q;

β j , B j

1,p

⊇ C μ

λ2

α j , A j

1,q;

β j , B j

1,p

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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 zqz/qz is starlike univalent in U and

R

1 α

γ qz 

zqz

qz −

zqz

qz

> 0, α, γ ∈ C, γ / 0. 3.1

If f ∈ A satisfies the subordination

α

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j

1,q;

β j , B j

1,p

fz

 γ

1z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz −zΦ

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j

1,q;

β j , B j

1,p

fz

≺ αqz γzqz

qz ,

3.2

then

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j

1,q;

β j , B j

1,p

Proof Our aim is to applyLemma 1.6 Setting

pz : z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j

1,q;

β j , B j

1,p

Computation shows that

zpz

pz 1 z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz − zΦ

I λ

α j , A j

1,q;

β j , B j

1,p

fz

ΦI λ

α j , A j

1,q;

β j , B j

1,p

fz , 3.5

which yields the following subordination:

αpz γzp

z

pz ≺ αqz γzqz

qz , α, γ ∈ C. 3.6

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By setting

θω : αω, φω : γ

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

Qz zqzφqz

 γz qz

qz ,

hz θ

qz

 Qz αqz γz qz

qz ,

3.8

we find that Qz is starlike univalent in U and that

R

zhz

Qz

1 α

γ qz 

zqz

qz −

zqz

qz

> 0. 3.9 Then the relation3.3 follows by an application ofLemma 1.6

Corollary 3.2 Let the assumptions of Theorem 2.1 hold Then the subordination

α − γ

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

 γ

1z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz

≺ αqz γzqz

qz ,

3.10

implies

z

I λ

α j , A j

1,q;

β j , B j

1,p fz

I λ

α j , A j

1,q;

β j , B j

1,p

Proof By letting Φω : ω.

Corollary 3.3 If f ∈ A and assume that 3.1 holds then

1 z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz ≺ 1 Az

1 Bz

A − Bz

implies

z

I λ

α j , A j

1,q;

β j , B j

1,p

fz

I λ

α j , A j

1,q;

β j , B j

1,p

fz ≺ 1 Az

1 Bz , −1 ≤ B < A ≤ 1, 3.13

and 1 Az/1 Bz is the best dominant.

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